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SEMICENTENNIAL  PUBLICATIONS 

OF   THE 

UNIVERSITY  OF  CALIFORNIA 


1868-1918 


THE  FUNDAMENTAL  EQUATIONS 

OF  DYNAMICS  AND  ITS  MAIN  COORDINATE 

SYSTEMS  VECTORIALLY  TREATED 

AND  ILLUSTRATED  FROM 

RIGID  DYNAMICS 


BY 

FREDERICK  SLATE 


UNIVERSITY  OF  CALIFORNIA  PRESS 

BERKELEY 

1918 


3SiG>0 


9 


THIS   BOOK   FORMS   PART   II   OF 

THE  PRINCIPLES   OF   MECHANICS,   PART  I, 

NEW  YORK,   THE   MACMILLAN   COMPANY,    1900 


PRESS  OF 

THE  NEW  ERA  PRrNTING  COMPANY 

LANOAS'ER,  PA. 

1918 


Sciences  Q  /\ 

Libranr       34  ^ 

5^3 


PREFACE 

The  day  has  clearly  passed  when  any  comprehensive  presen- 
tation of  all  dynamics  could  be  compressed  and  unijfied  within 
the  compass  of  one  moderate  volume  of  homogeneous  plan. 
The  established  connections  of  dynamical  reasoning  with  other 
fields  in  physics  are  of  increasing  number  and  closeness,  as 
furnishing  for  them  strongly  rooted  sequences  in  their  interpre- 
tative trains  of  thought  and  linking  together  what  would  else 
have  continued  to  stand  separate.  And  that  relation  has  reacted 
powerfully  in  modern  times  upon  dynamics  itself,  perpetually 
enriching  its  substance,  yet  at  the  same  time  introducing  within 
it  certain  sharpening  differences  that  are  stamped  upon  it  by  the 
type  of  use  for  which  preparation  is  being  made.  These  in  fact 
modify  superficially  the  modes  of  expression  and  their  tone,  and 
shift  their  own  emphasis  through  a  range  that  brings  about 
what  is  in  effect  a  subdivision  of  territory  and  an  acknowledg- 
ment of  practically  diverse  interests.  It  is  in  response  to  the 
situation  which  has  been  thus  unfolding,  and  in  conformity  with 
its  drift  toward  manifold  adaptations,  that  special  treatises  have 
been  rendered  available  whose  measure  of  unquestioned  excel- 
lence and  authority  would  make  superfluous  an  attempt  to 
replace  any  such  unit  with  a  marked  improvement  upon  it. 

But  undoubtedly  these  differentiations  founded  in  divergencies 
and  inevitably  expressing  them  in  some  degree,  are  entailing  a 
corresponding  need  and  demand  to  offset  them  with  a  broadening 
survey  of  the  common  foundation  and  of  the  common  stock  of 
resources.  And  with  that  end  in  view  another  treatment  of 
dynamics  finds  a  place  for  itself  and  holds  it  for  special  service. 
This  will  propose  to  state  with  catholic  inclusiveness  the  principles 


iv  Fundamental  Equations  of  Dynamics 

that  lay  out  and  direct  all  the  main  lines  of  use,  and  to  anticipate 
at  their  common  source,  as  it  were,  the  preferred  methods  and 
forms  that  are  characteristic  of  various  provinces. 

On  this  side  also  reasonable  requirements  for  the  immediate 
future  have  been  satisfied  up  to  a  definitely  recognizable  point. 
For  works  on  abstract  dynamics  are  at  hand  to  help,  whose 
number  and  quality  have  left  no  fair  opening  for  renewed  exposi- 
tion, that  could  indeed  scarcely  attain  excellence  without  dupli- 
cating them.  In  the  same  proportion,  however,  that  their 
requisite  perspective  has  grown,  until  it  involves  truly  panoramic 
sweep,  its  due  scope  must  cease  to  be  secured  except  from  a 
distance  that  expunges  most  details  and  spares  only  landmarks 
of  the  bolder  outlines.  And  under  the  urgent  pressure  to  con- 
dense in  order  to  avoid  neglecting  and  yet  not  become  too 
voluminous  in  summarizing  completely,  to  keep  even  pace  with 
widening  outlook,  this  view  of  dynamics  cannot  but  endure  the 
attendant  risks  of  abstractness.  Because  it  must  lean  in  building 
toward  great  rehance  upon  the  formal  aid  of  mathematics,  per- 
force the  physical  coloring  will  fade  and  the  bonds  with  experi- 
mental reasoning  be  loosened.  The  stated  results  are  pro- 
gressively less  likely  to  comprise  what  is  charged  with  tentative 
quality  and  is  held  with  the  candidly  provisional  acceptance 
proper  to  inductive  method. 

For  a  student  devoted  to  physical  science  though,  as  the 
gifted  mathematicians  Poincare  and  Maxwell  have  been  anxiously 
insistent  that  he  should  be  aware,  there  are  lurking  elements  of 
danger  in  magnifying  a  bare  logical  skeleton  as  a  goal,  and  in 
spending  best  effort  upon  articulating  it.  It  is  a  misguidance 
apt  to  control  into  rigidity  thought  which  can  scarcely  prove 
worthily  fruitful  unless  it  is  maintained  plastic.  There  is  a 
plain  sense  in  which  dependence  upon  clarity  of  demonstration 
in  terms  of  mathematical  brevity  and  rigor  may  operate  as  a 
defect;  and  that  severe  pruning  which  suppresses  all  but  defini- 


Introduction  v 

tive  advance  may  mislead.  There  is  a  season  for  mitigating  the 
austerity  of  algebra  and  daring  to  become  discursive,  for  relaxing 
the  ambition  that  is  steadied  to  attain  command  of  abstract 
principles  on  their  highest  level  and  for  pausing  in  reflective 
examination  of  their  genesis  and  their  setting.  Truly  it  would 
sterilize  action  to  inchne  thus  always;  but  never  to  turn  aside 
from  the  more  arduous  pursuit  tends  to  dissipate  that  atmosphere 
for  dynamics  which  has  given  it  life. 

At  the  other  extreme  are  found  the  practical  temperaments, 
looking  for  tools  with  which  to  undertake  their  special  tasks,  and 
largely  unmindful  of  the  processes  by  which  those  have  been 
shaped  and  of  the  far-reaching  equipment  in  which  their  func- 
tion is  but  one  part,  if  only  a  particular  routine  can  be  adequately 
served  or  intelligently  mastered.  And  this  more  empirical  frame 
of  mind  that  springs  from  absorption  in  monopohzing  pursuits 
can  be  fostered  and  strengthened  by  the  sheer  difliculties  in 
external  form  that  are  impressed  upon  abstract  dynamics  by  the 
tendencies  that  have  just  been  referred  to,  and  by  the  air  of 
remoteness  from  things  material  and  mundane  which  that 
treatment,  if  uncorrected,  confers.  Unless  it  can  be  halted, 
therefore,  a  movement  toward  disintegration  which  must  be 
coped  with  will  confront  the  cultivators  of  dynamics  that 
derives  a  backing  also  from  other  circumstances  of  the  present 
situation. 

The  Hfting  of  technical  science  to  a  better  plane,  where  the 
habitual  facing  of  new  problems  under  the  illumination  of 
theoretical  insight  is  coming  to  prevail,  creates  a  demand  in  all 
the  fundamental  sciences  that  is  a  modern  appeal.  It  has  been 
incorporated  into  fixed  plans  of  preparation  for  normal  careers 
in  active  life  to  accompHsh  those  things  which  were  formerly 
undertaken  with  dominating  incHnation  by  minds  self-selected 
through  their  special  gifts.  There  must  be,  then,  in  the  methods 
of  presentation  and  in  the  execution  of  them,  some  recognition 


vi  Fundamental  Equations  of  Dynamics 

of  a  constituency  that  is  at  once  larger,  less  homogeneous,  and 
more  in  need  of  aid.  In  a  restricted  sense  of  the  word,  there  is 
a  summons  to  popularize  the  abstruser  sciences,  and  among  them 
dynamics,  with  a  design  to  favor  their  assimilation  by  students 
at  an  earher  stage.  This  will  make  concessions  in  view  of 
hindrances  inherent  in  the  subject-matter,  and  allowance  for 
faculties  of  comparison  and  of  analytic  judgment  not  yet  ripened 
into  full  command  of  all  resources. 

There  is  some  element  in  the  immediate  need  that  is  due  to 
passing  a  transition  and  that  will  be  lost  in  a  newly  adjusted 
order;  for  it  has  appeared  from  manifold  experience  what 
marvels  can  be  wrought  by  tradition  in  giving  easy  currency  to 
scientific  doctrine.  Moreover,  the  obstacles  that  loomed  larger 
by  mere  novelty  suffer  genuine  reduction  by  more  lucid  state- 
ment. An  older  generation  arrived  but  gradually  at  an  under- 
standing of  the  principle  in  conservation  of  energy,  and  caught 
the  advantage  and  power  of  absolute  measurements  first  in 
glimpses.  Yet  they  have  lived  to  find  those  unfamiliar  ideas 
adopted  among  the  smoothly  working  formulas  of  unquestioned 
truth.  So  it  will  not  pass  the  limits  of  a  reasonable  anticipation 
to  forecast  how  the  younger  generation  of  today  can  move  at 
ease  in  their  maturity  among  bold  concepts  that  were  obscure 
when  imperfectly  grasped.  Nevertheless,  as  the  call  now  is, 
so  must  the  answer  be  given. 

Every  aspect  of  the  thoughts  here  put  down  is  framed  in  a 
personal  experience:  the  profit  from  quickening  perception  and 
appreciation  for  the  nexus  between  sharply  generalized  ideas 
and  their  narrower  origins;  the  benefit  of  laying  stepping- 
stones  gauged  to  a  student's  stride;  the  reward  of  implanting 
human  interest  within  the  routine  of  an  industrial  calling;  also 
the  moral  gain  through  confirming  intellectual  honesty  under  a 
sustained  demand  for  actual  comprehension  of  what  one  is 
challenged  to  attack  among  the  papers  rated  as  classics,  or  in 


Introduction  vii 

judging  and  sifting  recent  work.  Aiding  to  scent  difficulties 
first  and  then  to  overcome  them  fits  the  processes  of  the  average 
mind,  where  the  stronger  talent  can  walk  self-guided. 

The  present  enterprise  was  born  of  the  foregoing  considerations 
in  so  far  as  they  dictated  its  material  and  the  ends  for  which  that 
was  offered  in  gradual  accumulation  during  many  years  and 
under  the  influence  of  contact  with  students  of  varied  purpose. 
It  renounces  from  the  outset  all  claim  to  be  systematically  con- 
ceived; it  is  content  with  a  circling  return  from  one  point  and 
another  to  a  core  of  ideas  that  are  worth  reviewing  in  their 
various  aspects  because  they  are  central.  In  their  nature  being 
a  supplement  to  standard  books  that  differ  in  type  from  each 
other,  and  offering  themselves  in  flexible  continuation  of  an 
elementary  stage  with  unsettled  achievement,  these  selected  dis- 
cussions cannot  escape  being  judged  fragmentary  by  some,  redun- 
dant by  others.  But  their  spirit  and  their  general  aim  are  built 
upon  ascertained  failure  to  acquire  elsewhere  a  just  comprehen- 
sion of  several  matters  here  made  prominent  and  perhaps  in 
some  degree  originally  presented. 

This  kernel  of  intention  in  the  subject-matter  gathered  for 
these  chapters  lends  to  them,  it  may  be  claimed  legitimately, 
something  of  peculiar  appropriateness  for  the  circumstances  of 
their  publication.  On  the  occasion  to  be  celebrated  it  seems 
particularly  pertinent  that  there  should  be  recorded  in  some 
permanent  form  the  working  of  those  influences  which  our 
University  has  not  withheld  from  her  graduates,  to  nourish  in 
them  a  living  root  of  independent  thinking  and  of  unflinching 
thoroughness  without  which  constructive  scholarship  cannot 
exist. 

June,  1917. 


CONTENTS 
CHAPTER  I 

PAcn 

Introductory  Summary 1 

CHAPTER  II 
The  Fundamental  Equations 21 

CHAPTER  III 
Reference  Frames  :   Transfer  and  Invariant  Shift  ...     76 

CHAPTER  IV 

Some  Coordinate  Systems 112 

Notes  to  Chapters  I-IV 201 

Index 225 


CHAPTER   I 

Introductory  Summary 

1.  Only  sciences  that  have  attained  a  certain  ripeness  of 
strongly  rooted  development  have  been  found  capable  of  com- 
bining a  vigorous  and  progressive  activity  at  their  working 
frontier  for  advance  with  reflective  examination  of  their  deeper 
foundations  and  their  general  method.  The  activity  is  aggressive 
in  devising  novel  attack  upon  enlarging  material,  while  reflection 
upon  what  has  already  become  standard  must  go  with  recasting 
it  to  meet  modified  demands.  This  situation  has  been  promi- 
nently reaUzed  in  the  case  of  dynamics,  whose  stirrings  to  self- 
criticism  have  been  evermore  spurred  by  the  interactions  with 
mathematics  and  astronomy',  its  closer  neighbors,  at  the  same 
time  that  its  field  was  broadening  to  permeate  and  harmonize  the 
greater  part  of  physics.  A  large  net  gain  of  helpful  stimulus  from 
common  aim  must  be  allowed  here,  reenforcing  the  vigor  from 
rapid  growth,  though  there  have  been  some  dangers  for  dynamics 
to  avoid,  such  as  becoming  infected  with  the  more  formal  and 
abstract  spirit  of  mathematics,  or  underrating  its  own  basis  in 
phenomena  by  acquiescing  too  generously  in  philosophy's  rating 
for  empirical  science.  It  is  a  fitting  preliminary  to  our  immediate 
purpose  to  touch  upon  one  or  two  such  reactions  between  in- 
fluences from  without  and  from  within;  in  part  because  the 
inquiries  that  were  provoked,  though  prolonged  through  fifty 
years  or  more  with  acuteness  and  tenacity,  have  left  practically 
unshaken  the  external  forms  of  quantitative  expression,  at  least. 
This  is  no  sign,  however,  that  djmamics  is  stationary  and  stereo- 
typed; but  onlj^  a  reassuring  fact  to  beget  confidence  in  the 
fabric  of  the  science.     The  subtle  and  less  obtrusive  changes 

1 


2  Fundamental  Equations  of  Dynamics 

must  not  be  forgotten,  that  have  clarified  the  concepts  and 
infused  into  them  added  significance  by  revised  interpretation. 
Reading  the  prospects  of  the  imminent  future,  too,  rouses  the 
expectation  that  what  has  been  will  continue  to  be,  while  dy- 
namics is  adapting  itself  to  a  wider  scheme  of  connections  and 
to  a  more  accurate  insight  into  its  own  doctrine  or  theory. 

It  is  indeed  an  astonishing  testimony  to  the  happy  strokes  of 
genius  in  the  founders  of  mechanics  that  force,  impulse,  work, 
momentum  and  kinetic  energy  still  exhaust  the  primary  needs, 
though  the  broader  scope  of  dynamics  now  covers  the  chain  of 
transformations  in  which  mechanical  energy  is  only  one  link. 
And  it  confirms  our  belief  in  the  vital  and  definitive  appropriate- 
ness of  those  quantities  to  find  them  retained  essentially  by  those 
who  are  trying  out  another  body  of  principles  that  might  be  substi- 
tuted entirely  or  in  part  for  the  Newtonian  mechanics.  Mean- 
while the  equations  of  motion  have  not  been  superseded,  yet 
they  date  from  the  seventeenth  century;  the  notable  advances 
due  to  d'Alembert,  Euler  and  Lagrange  in  the  eighteenth  century, 
and  to  Hamilton  in  1835,  offer  still  the  foundations  upon  which 
we  build.  But  this  introduction  would  outline  a  one-sided  and 
misleading  picture  of  mere  static  stability  unless  it  used  its 
warrant  in  bringing  to  supplementary  notice  three  strands  that 
have  been  woven  into  dynamics  more  recently,  to  alter  in  some 
degree  its  texture  and  to  influence  its  emphasis.  We  shall  next 
attempt  to  dispose  of  these  in  all  proper  brevity. 

2.  Under  the  first  label  energetics  we  are  called  upon  to  chron- 
icle a  strong  movement  that  sought  to  enhance  the  prestige  that 
energy  in  its  various  forms  had  already  gained  by  the  rapidly 
successful  campaign  about  the  middle  of  the  nineteenth  century.^ 
This  tendency  was  an  almost  inevitable  accompaniment  of  that 
dominating  relation  to  physical  processes  which  conservation  of 
energy  as  a  conceded  central  principle  had  justified  beyond  cavil. 

1  See  Note  1.     Refer  to  collected  notes  following  Chapter  IV. 


Introductory  Summary  3 

But  the  more  pronounced  utterances  about  energy  overshot  the 
mark  in  their  zeal,  and  sought  to  exalt  it  in  rank  as  the  one 
dynamical  quantity  to  which  the  rest  should  be  held  auxiliary, 
and  upon  which  they  should  be  based  mathematically.  Then 
the  series — kinetic  energy,  momentum,  force,  mass — was  to  be 
unfolded  out  of  its  first  term  by  divisions;  and  violent  extremists 
were  heard,  even  condemning  force  as  a  superfluous  concept, 
refusing  to  associate  it  directly  with  our  muscular  sense,  or  to 
recognize  it  as  an  alternative  point  of  departure  yielding  momen- 
tum and  other  quantities  by  multiplications.  Of  course  deliber- 
ate minds  looked  askance  at  a  professedly  universal  point  of 
view  that  would  exclude,  save  at  the  cost  of  an  artificial  device, 
such  important  elements  as  constraints  that  do  no  work.  Com- 
mon sense  declined  to  cripple  our  assault  upon  problems  for 
doctrinaire  reasons  that  would  bar  and  mark  for  disuse  certain 
highways  of  approach,  but  it  seized  the  chance  instead  to  enrich 
and  strengthen  dynamics  by  wisely  adopting  the  suggestion  to 
exploit  more  completely  the  relations  that  energy  specially 
furnishes,  and  to  incorporate  them  among  its  resources  and 
methods.  After  abating  its  flare  of  exuberance,  the  saner  forces 
behind  the  reconstruction  that  was  advocated  have  been  har- 
nessed and  made  contributory  to  a  real  advance  that  grafts  new 
upon  old,  and  embraces  whatever  proved  advantage  attaches  to 
all  reasonable  points  of  view,  with  the  object  of  reducing  finally 
their  oppositions  and  fitting  them  in  place  within  a  more  compre- 
hensive survey. 

What  is  patent  to  read  in  the  example  of  energetics  should  in 
prudence  be  made  further  to  bear  fruit;  since  judging  historically, 
any  new  burst  of  reform  spirit  will  be  likely  to  repeat  the  main 
features  of  its  lesson.  An  old  and  thoroughly  tested  science 
especially  will  less  easily  break  the  continuity  of  its  course, 
though  it  is  always  responsively  ready  to  swerve  under  every 
fresh   impulse  to   amendment  by  discovery.     So  the  matters 


4  Fundamental  Equations  of  Dynamics 

offered  recently  under  the  caption  relativity  are  surely  giving  to 
dynamics  a  wider  sweep  of  horizon;  but  there  too,  when  the 
permanent  benefit  accruing  has  been  sifted  out,  the  residue  will 
probably  prove  more  moderate  than  the  tone  of  radical  spokes- 
men has  been  implying  while  the  sensation  of  novelty  was 
strongest.^ 

3.  It  has  been  remarked  often  that  Newton's  three  laws  of 
motion  taken  by  themselves  give  a  bias  toward  concentrating 
attention  upon  momentum,  and  upon  force  exclusively  as  its 
time-derivative,  with  a  comparative  neglect  of  the  counterpart 
in  work  and  its  relation  to  force.  The  restoration  of  balance 
began  at  once  however,  and  soon  the  principle  of  vis  viva  was 
added  and  recognized  as  complementary  on  a  level  footing  to 
Newton's  second  law.  The  equivalents  of  what  are  now  known 
as  the  impulse  equation  and  the  work  equation  were  established 
firmly  and  put  to  use.  The  readjustment  thus  begun  was 
continued  by  steps  as  their  desirableness  was  felt  until  with  the 
ripeness  of  time  it  culminated,  we  may  say,  in  the  proposals 
that  form  the  nucleus  of  what  we  call  energetics.  It  will  be 
profitable  to  expand  that  thought  and  mention  some  chief 
sources  of  the  need  to  follow  that  line,  or  what  gain  has  been 
found  in  doing  so. 

In  rudimentary  shape  the  idea  of  conservation  of  energy  had 
emerged  early;  the  histories  are  apt  to  date  it  from  the  method 
invented  by  Huyghens  for  the  treatment  of  the  pendulum. 
And  so  soon  as  the  formal  step  had  been  taken  in  addition,  that 
set  apart  under  the  heading  potential  energy  the  work  of  weight 
and  of  gravitation,  because  it  can  be  anticipated  by  advance 
calculation  exactly  and  with  full  security,  the  invariance  of 
mechanical  energy  under  the  play  of  these  forces  when  thus 
expressed,  or  its  conservation  within  these  narrower  hmits, 
became  a  demonstrable   corollary  of  fundamental   definitions. 

1  See  Note  2. 


Introductory  Summary  5 

The  discovered  inclusion  of  electric  and  magnetic  attractions  or 
repulsions  under  the  same  differentially  applied  law  of  inverse 
square  that  is  characteristic  of  gravitation  made  natural  the 
extension  of  potential  energy  as  a  statement  of  securely  antici- 
pated work  to  the  field  of  those  actions  as  well.  And  a  large 
group  of  valuable  mathematical  consequences  was  accumulated 
which  remain  classic  and  which  accompany  the  law  of  inverse 
square  wherever  it  may  lead,  retaining  their  validity  with  only 
slight  changes  of  detail. 

These  developments  are  controlled  to  a  great  extent  by  the 
idea  of  energy,  and  they  must  have  built  up  a  general  perception 
of  its  power.  The  invariance  of  energy  was  fitted  more  com- 
pletely for  use  as  a  principle,  wherever  its  mechanical  forms  alone 
enter  which  we  distinguish  as  kinetic  and  potential,  when  Gauss 
had  evolved  that  plan  of  so-called  absolute  measure  which  has 
furnished  us  with  the  centimeter-gram-second  system.  He 
certainly  consolidated  into  unity  all  sources  of  ponderomotive 
force  in  the  several  fields  where  a  potential  had  been  recognized. 
Of  course  we  discriminate  between  this  stage  and  the  conserva- 
tion of  energy  under  all  its  transformations  to  which  the  period 
of  Mayer,  Joule  and  their  coworkers  attained.  The  earlier 
halting-place  behind  distinct  limitations  of  scope  left  matters 
besides  with  a  formal  content  only,  in  the  sense  that  no  questions 
were  raised  and  squarely  faced  that  looked  toward  localizing  the 
latent  energy  and  investigating  the  possible  mechanism  by  which 
a  medium  might  hold  it  in  storage.  This  formal  mathematics 
centered  on  the  fact  that  the  work  done  within  a  conservative 
system  and  between  the  same  terminal  configurations  does  not 
depend  upon  the  particular  paths  connecting  them.  It  is  a 
strikingly  significant  exhibition  of  that  quasi-neutrality  that  is 
now  one  salient  and  accepted  feature  in  the  procedure  of  ener- 
getics that  so  much  of  solid  and  permanent  accomplishment  was 
possible  while  certain  vital  issues  were  evaded,  and  without 


6  Fundamental  Equations  of  Dynamics 

being  compelled  to  register  even  a  tentative  decision  upon  them. 
That  non-committal  attitude  towards  much  else  as  subsidiarj', 
provided  always  that  the  gains  and  losses  of  energy  for  the 
system  under  consideration  can  be  made  to  balance,  has  often 
been  employed  to  turn  the  flank  of  obstacles  and  has  been  in  that 
respect  an  element  of  strength.  Or  it  leaves  us  in  the  lurch 
weakly,  we  might  say  about  other  occasions  where  we  have 
stood  in  need  of  some  crucial  test  between  alternatives,  and  have 
found  but  a  dumb  oracle. 

4.  The  next  important  advance  was  then  timely  and  specially 
fruitful  in  giving  life  and  deeper  meaning  to  what  had  been  in 
these  directions  more  a  superficial  form;  and  at  the  same  time 
in  moving  forward  beyond  the  previous  stopping-place  to  expand 
the  range  of  dynamical  ideas.^  It  is  Maxwell  who  is  credited  with 
initiating  these  contributions  by  treating  dynamically  new 
aspects  of  electromagnetic  phenomena.  He  took  bold  and  novel 
ground  by  outlining  his  provisional  basis  for  an  electromagnetic 
theory  of  light  that  converted  a  colorless  temporary  vanishing  of 
energy  into  a  definite  and  plausible  plan  for  its  storage  in  a 
medium.  In  achieving  this  change  of  front  he  brought  three 
lines  of  thought  to  a  convergence-point;  for  besides  the  re- 
searches of  Faraday  and  those  that  identified  quantitatively  the 
many  transformations  of  energy,  he  utilized  more  fully  than 
his  predecessors  had  dared  the  possibilities  that  the  earlier 
dynamics  had  done  much  toward  making  ready  to  his  hand. 
It  is  this  third  element  perhaps  that  marks  most  strongly  for 
us  the  threshold  of  the  new  enterprise  upon  which  dynamics  will 
hereafter  be  engaged,  in  whose  tasks  we  can  find  a  union  in  just 
proportion  of  imaginative  speculation  with  mastery  of  the 
mathematical  instruments  and  with  the  candid  policy  of  ener- 
getics to  preserve  an  open  mind  and  a  suspended  judgment  in 
the  face  of  undecided  questions. 

1  See  Note  3. 


Introductory  Summary  7 

Maxwell  was  a  pioneer  in  prolonging  with  new  purpose  the 
sequence  upon  which  d'Alembert  set  out,  and  which  Lagrange 
continued,  beyond  the  point  at  which  the  latter  paused  after 
recording  notable  progress.  What  those  earlier  men  had  done 
with  the  discovery  of  virtual  work  as  a  basis  for  developing 
mechanics  remained  to  be  restated  for  dynamics,  and  adapted 
to  a  more  inclusive  command  of  energy  transformations.  Among 
other  things  this  has  given  us  an  enlarged  interpretation  of  older 
terms.  We  are  ready  to  view  a  conservative  sj'^stem  as  one 
whose  energy  processes  are  reversible:  that  is,  energy  of  any 
form  being  put  in,  it  can  be  restored  without  loss,  in  the  same 
form  or  in  some  other.  We  have  learned  to  group  fair  analogues 
of  kinetic  and  of  potential  energy  for  a  system  thus  conservative 
according  to  one  defensible  test.  Potential  forms  of  energy  will 
be  found  resilient  as  the  original  examples  are;  that  is,  they  will 
exhaust  themselves  automatically,  under  the  conditions  of  the 
particular  combination,  unless  the  corresponding  transformation 
is  prevented  actively.  But  in  order  to  be  coordinated  with 
kinetic  energy  on  the  other  hand,  the  passive  quality  must  be  in 
evidence  that  requires  some  decisive  intervention  for  the  passage 
into  other  forms.  This  trend  toward  assigning  wider  meaning 
to  dynamical  concepts  has  given  us  further  generalized  force  as  a 
quotient  of  energy  by  a  change  in  its  correlated  coordinate;  the 
matching  of  force  and  coordinate  as  factors  in  the  product  that 
is  energy  being  executed  on  due  physical  grounds.  We  have 
been  led  likewise  to  replace  mass  by  a  broader  term  inertia, 
where  a  quantity  is  detectable  in  the  phenomena  of  more  general 
energy-storage,  that  stands  in  essential  parallelism  with  the  rela- 
tion of  mass  itself  to  force  and  kinetic  energy.  And  the  dynami- 
cal scheme  has  been  rounded  out  by  allowing  to  momentum  those 
privileges  of  latency  and  of  reappearance  in  the  literal  mechanical 
form,  that  were  at  the  outset  the  monopoly  of  energy. 

5.  These  comments  have  been  attached  to  Lagrange's  equa- 

2 


8  Fundamental  Equations  of  Dynamics 

tions  because  Maxwell  did  in  fact  make  them  the  vehicle  of  his 
thought;  insisting  upon  sufficient  detail  to  lift  the  reproach  of 
indefiniteness,  but  also  by  a  right  inherent  in  the  method  passing 
over  in  silence  the  points  where  invention  had  thus  far  failed.  But 
it  was  demonstrated  long  ago  that  d'Alembert  and  Lagrange  and 
Hamilton  have  provided  us  with  interconnected  lines  of  approach 
to  the  same  goal;  except  as  the  element  of  choice  is  directed  by 
convenience  Hamilton's  principle  lends  equal  favor  and  support 
with  Lagrange's  equations  to  the  attempt  to  summarize  a  com- 
prehensive statement  in  terms  of  energy.  The  former  however 
elects  to  generalize  for  all  analogous  transformations  upon  a 
simple  theorem:  That  potential  energy  will  exhaust  itself  as 
rapidly  as  imposed  constraints  allow  upon  producing  kinetic 
energy. 

Beside  the  direct  intention  to  indicate  some  reasons  why 
dynamics  leans  increasingly  upon  energy  relations,  and  borrows 
from  energetics  some  modes  of  attack,  these  later  remarks  have 
a  reverse  implication  as  well.  They  intimate  the  belief  that 
firm  hold  upon  the  elementary  content  of  dynamical  principles 
and  intelligent  full  insight  into  them  are  not  superseded,  nor 
yet  to  be  shghted.  And  the  meaning  here  is  not  the  mere  com- 
monplace truth  that  the  more  modest  range  satisfies  many  needs; 
or  that  historically  it  is  the  tap-root  that  has  nourished  and 
sustained  the  later  growth.  But  recurring  to  what  lies  at  the 
foundation  is  further  the  best  preparation  for  the  critical  dis- 
crimination that  must  be  exercised  at  the  advancing  frontier, 
because  it  holds  the  clews  of  conscious  intention  by  which  all 
effort  there  has  been  guided,  and  lends  effective  aid  in  steering 
an  undeflected  course  among  a  medley  of  proposals  to  tolerate  in 
concepts  a  figurative  shading  of  their  literal  acceptation,  or  to 
condone  acknowledged  fictions  on  grounds  of  expediency. 

6.  The  redistribution  of  emphasis  upon  which  we  have  been 
dwelling  has  doubtless  exercised  the  most  penetrating  influence 


Introductory  Summary  9 

to  alter  the  complexion  of  mechanics  as  Newton  left  it,  and 
therefore  we  have  put  it  first.  But  there  has  been  a  second 
movement  whose  modifying  effect  as  dynamics  has  grown  must 
not  be  neglected,  and  which  also  like  the  leavening  with  energetics 
has  been  spread  over  a  considerable  period,  though  our  report 
of  its  outcome  can  be  compressed  into  a  brief  space. ^  This 
exhibited  itself  in  a  searching  and  protracted  discussion  on  the 
relativeness  of  velocity  and  acceleration  that  did  its  part  in  con- 
tributing to  clearness  by  removing  ambiguity  from  a  group  of 
terms  and  carrying  through  a  completer  analysis  of  their  bearings. 
The  main  concern  here  was  not  so  much  with  the  baldly  kine- 
matical  side  of  the  question;  since  it  is  plain  that  the  final  truth 
in  that  sense  lies  very  near  the  surface.  But  the  endeavor  was 
quite  specially  shaped  by  the  ambition  to  contrive  at  least  soundly 
consistent  expression  for  all  dynamical  processes  that  shall  be 
recognized  in  physics;  perhaps  with  some  reach  toward  an  ideal 
of  universal  and  ultimate  validity.  The  entire  relativeness  of 
those  motions,  which  furnish  leading  factors  of  importance  in 
decisions  upon  working  values  of  dynamical  quantities,  is  now  a 
standard  item  in  the  opening  chapters  of  dynamics  as  a  corol- 
lary to  choice  of  reference  elements  by  agreement. 

The  acquirement  of  this  point  of  view  has  therefore  excluded 
all  search  for  truly  absolute  motion  and  canceled  the  unqualified 
significance  of  the  phrase  which  dates  as  far  back  as  Newton. 
Since  it  seems  flatly  contradictory  to  unshackled  relativeness, 
an  impression  may  be  created  at  first  hearing  that  here  for  once 
the  older  thought  has  been  overturned  and  radically  revised. 
Yet  the  case  is  not  so  weak  as  it  sounds,  nor  do  we  see,  when 
we  look  below  the  surface,  that  any  foundations  have  been 
affected  vitally.  We  may  be  comforted  to  observe  only  another 
striking  instance  where  a  great  mind  did  not  everywhere  and 
straightway  hit  upon  most  felicitous  terms  to  describe  how  it 

•  See  Note  4. 


10  Fundamental  Equations  of  Dynamics 

dealt  with  powerful  nascent  conceptions.  Newton  seems  to  call 
motions  absolute  if  they  dovetailed  easily  with  the  spacious 
frame  of  physical  action  that  his  discovery  of  gravitation  was 
beginning  to  build;  and  himself  engrossed  in  a  swift  recon- 
naissance through  the  new  region,  he  left  later  invention  to 
amend  his  notation.  But  it  is  chiefly  the  philosophical  conno- 
tations of  his  word  absolute  and  not  its  unfitness  in  physics  that 
have  made  it  the  center  of  futile  controversy.  Thus  the  idea  that 
the  older  writers  really  had  in  mind  when  they  spoke  of  absolute 
motion  was  scarcely  different  from  one  that  continues  to  hold 
its  ground  and  compels  us  still  to  separate  two  lines  of  inquiry. 
Because  beyond  the  settlement  of  kinematical  equivalences  that 
is  direct  and  simple  since  it  is  unhampered  by  any  physical 
considerations,  the  questions  of  real  difficulty  remain  unsettled 
to  confront  us.  They  have  had  a  certain  elusive  character  by 
involving  a  complicated  and  tentative  estimate  that  must 
balance  on  the  largest  scale  and  through  the  whole  range  of 
physics  net  gain  against  loss  in  simplicity.  What  common  back- 
ground, as  it  were,  of  reference-elements  is  decipherable  upon 
which  the  interplay  of  forces  and  of  energies  shall  stand  in 
simplest  and  most  consistently  detailed  relief? 

In  consequence  it  has  not  been  displaced  as  a  tenet  of  orthodox 
dynamical  doctrine  that  standards  by  which  to  judge  of  the 
energy,  momentum  and  force  that  ought  to  appear  in  its  accounts 
will  not  stand  on  a  par  if  adopted  at  random,  however  inter- 
changeable they  have  proved  in  passing  upon  rest,  velocity  and 
acceleration  by  the  mathematical  criteria  in  the  more  indifferent 
domain  of  kinematics.  Dynamics  has  never  hesitated  to  stig- 
matize apparent  forces,  for  example,  as  spurious  or  fictitious  in 
relation  to  its  general  procedure,  and  to  revise  its  lists  of  rejec- 
tions on  due  grounds  derived  from  advance  in  knowledge  and 
in  method.  The  definitive  resolution  of  uncertainties  that  affect 
reasonable  decision  for  the  questions  here  implied  is  still  awaited; 


Introductory  Summary  11 

of  necessity  that  objective  is  not  attainable  conclusively  while 
the  surveys  in  the  several  provinces  of  physics  remain  both 
fragmentary  and  disconnected.  Though  it  has  been  claimed 
indeed  that  secure  foothold  was  being  gained  through  reliance 
upon  a  reference  to  stellar  arrangement  in  removing  excres- 
cences that  showed  by  the  light  of  its  corrective  tests. 

7.  The  growing  practice  to  designate  that  reference  as  ultimate, 
however,  has  not  excluded  a  proper  admission  that  its  lines  of 
specification  were  to  be  improved  by  whatever  greater  precision 
new  discovery  and  analysis  of  it  reveal  definitely  to  be  progress. 
And  it  is  fairly  probable  that  majority  opinion  was  looking 
entirely  in  that  direction  for  fresh  landmarks  until  other  prospects 
were  opened  with  vigor  in  recent  years.  These  depend  upon  a 
certain  increase  in  freedom  to  retain  functional  forms  when  the 
time-variable  is  added  to  the  coordinates  and  included  in  the 
group  of  quantities  that  are  involved  in  the  readjustment  when  a 
change  of  base  in  the  reference  is  undertaken.  This  far-reaching 
proposal  derived  its  original  suggestion  from  optical  phenomena 
peculiar  to  electromagnetism  and  in  one  sense  exceptional;  yet 
since  it  is  the  crux  of  this  situation  that  a  decision  of  universal 
application  is  sought,  any  unreconciled  indications  of  alternative 
must  be  reckoned  with,  whereby  two  plans  for  attaining  the 
maximum  simplicity  that  is  desired  become  divergent.  The 
competitive  schemes  of  ultimate  reference  cannot  be  weighed 
decisively  before  the  ramifications  of  both  have  been  traced 
everywhere  in  that  detail  which  can  afford  a  satisfactory  con- 
clusion through  their  final  comparison.  And  for  that  the  time 
does  not  seem  ripe;  especially  as  each  thus  far  falls  short  of 
established  universal  quality  by  seeming  to  leave  some  combina- 
tions unreduced,  or  abnormal  to  its  plan.  It  is  therefore  reassur- 
ing to  our  logical  sense  to  note  how  the  practically  available 
devices  of  proximate  reference  persist  and  are  neutral,  save  in 
the  formulation  of  the  Hmits  upon  which  their  steps  of  increasing 


12  Fundamental  Equations  of  Dynamics 

precision  may  be  declared  to  converge.  For  that  their  own 
framework  is  by  spontaneous  intention  approximate  can  be 
conceded  without  discussion. 

8.  The  contrast  upon  which  we  have  been  remarking,  between 
an  indecision  toward  many-phased  equivalences  and  the  evolution 
of  preference  among  them  is  then  one  characteristic  of  the  trans- 
ition from  kinematics  to  dynamics;  that  is,  from  a  range  fixed  by 
mathematical  conformity  to  a  selection  narrowed  by  physical 
meanings.  We  can  proclaim  a  forward  step  in  that  direction  when 
the  allowable  mathematical  range  has  been  plausibly  dehmited,  as 
with  the  transverse  wave  of  optics  from  Fresnel's  wave-surface  in 
crystals  to  recent  descriptive  spectroscopy;  but  it  is  the  crown  of 
attainment  to  master  insight  into  the  causes  of  the  effects  ob- 
served, or  into  their  sources,  or  into  their  explanations,  in  whatever 
chosen  terms  the  phrase  may  stand.  This  persistent  effort  to 
identify  physical  sequences  with  a  mechanism,  to  link  a  series 
of  phenomena  by  means  of  a  mechanical  interpretation,  has 
absorbed  its  full  quota  of  sanguine  activity  since  Newton  scored 
his  early  partial  success  with  the  propagation  of  sound.  The 
record  shows  in  the  main  that  the  harvest  of  reward  for  these 
attempts  has  continued  into  this  later  era,  slackening  somewhat 
of  course  by  exhaustion  of  the  material.  Yet  there  has  been, 
too,  a  baffling  of  the  imagination  in  its  task  of  dissecting  the 
complicated  workings  of  energy  in  less  traceable  manifestations 
by  traveling  on  parallels  to  direct  sense-experience.  And  again 
optics  illustrates;  but  now  is  shown  a  kind  of  failure,  both  with 
the  abandoned  types  of  its  theory  and  in  its  electromagnetic 
alliance. 

Every  move  in  bestowing  thus  upon  dynamics  the  control  of  a 
larger  domain  has  been  healthy  growth,  keeping  pace  with 
progress  in  other  directions;  and  always  sufficiently  safeguarded 
against  speculative  vagueness  by  bonds  with  the  method  of  its 
beginnings.     Wherever  mechanical  energy  in  ponderable  masses 


Introductory  Summary  13 

exhibits  itself  in  the  actual  chain  of  transformations,  it  gives  a 
touchstone  through  the  measurable  quantities,  Uke  pondero- 
motive  force,  by  which  to  try  the  conceived  series  for  its  vahdity 
or  consistency. 

There  are  assumed  successions,  however,  in  which  mechanical 
energy  is  not  directly  in  evidence  though  equivalents  of  it  appear 
in  amounts  known  by  using  the  change-ratios.  Suppose  we  trans- 
late the  given  facts  or  quantities  and  introduce  mechanical  energy 
fictitiously.  We  have  been  prone  to  incline  our  judgment  of  the 
original  case  according  to  the  analogies  of  its  artificial  substitute, 
and  accordingly  to  accept  the  assumptions  of  the  former  or  to 
speak  skeptically  of  its  paradoxes.  But  in  the  puzzHng  region 
that  we  have  just  mentioned  there  may  be  written  a  hidden 
caution  about  the  cogency  of  such  transferred  conclusions.  The 
absence  of  mechanical  energy  from  the  transformations  that  do 
occur,  as  we  are  ready  to  suppose  for  light  during  transmission, 
or  for  a  free  electron  with  inertia  and  without  mass  but  traversing 
an  electromagnetic  field,  may  be  a  contributory  circumstance  in 
precluding  a  mechanical  model  and  in  leaving  us  thus  far  in  the 
twilight  of  kinematics,  wrecked  on  obstacles  of  seeming  internal 
contradiction.  And  to  the  extent  to  which  this  indicated  possi- 
bility is  entertained,  the  leverage  of  these  unreduced  phenomena 
will  be  diminished,  to  guide  or  to  modify  dynamical  thought  that 
discusses  ponderable  materials. 

9.  The  third  gain  that  we  must  bring  forward  is  the  improved 
formulation  of  dynamics  by  replacing  the  cartesian  expansions 
with  vector  analysis,  whenever  general  discussions  and  theorems 
are  taken  in  hand,  or  indeed  everywhere  unless  we  are  barred 
by  the  needs  of  detailed  calculation  to  which  the  vector  notation 
is  not  so  well  adapted.  The  direct  influence  here  is  confined  to 
external  forms,  it  is  true;  yet  indirectly  an  undeniable  effect 
will  always  be  exerted  to  favor  continuity  in  the  presentation  of 
reasoning,   and   to   preserve  with  fewer  breaks  an  intelligent 


14  Fundamental  Equations  of  Dynamics 

orientation  during  extended  developments.  These  advantages 
are  felt  already,  and  they  will  accrue  perpetually  as  a  natural 
accompaniment  of  increased  compactness  in  stating  relations 
and  of  accentuating  resultants  first,  only  passing  on  to  their 
partial  aspects  where  necessary.  We  should  all  lend  our  aid  to 
banish  the  obscurities  and  the  disguises  inseparable  from  the 
older  system  of  equation-triplets.  The  subdivision  of  the  newer 
analysis  that  is  known  distinctively  as  vector  algebra  is  stand- 
ardized fairly  to  the  point  of  rendering  great  help  in  dynamics, 
and  adjustments  to  this  specific  use  are  perfecting.  As  regards 
the  vector  operators  like  gradient,  curl  and  divergence,  they  are 
as  yet  far  from  establishment  in  full  effectiveness,  by  unforced 
extension  of  their  original  relation  to  field-actions  and  abatement 
of  its  comparative  abstruseness. 

10.  This  introduction  will  distort  the  truth  of  its  own  words 
and  convey  an  unbalanced  false  impression,  unless  our  reading 
of  it  can  be  depended  upon  to  counterpoise  the  omissions  that  have 
trimmed  it  to  these  succinct  proportions.  So  it  is  well  to  make 
room  at  this  point  for  a  few  sentences  that  bear  upon  maintaining  a 
real  perspective  against  the  tendency  of  extreme  compression. 
And  first  it  must  be  realized  that  the  personal  careers  of  a  small 
group  of  geniuses  do  not  constitute  scientific  history.  To  men- 
tion one  great  man  and  to  picture  him  advancing  with  long  sure 
strides  implies  with  scarce  an  exception  a  whole  accompanying 
period.active  with  sporadic  anticipations  of  some  larger  swing;  an 
epoch  of  transition  busy  with  foreshadowings  of  a  new  alignment. 
One's  own  thought  should  always  supply  this  current  of  perhaps 
unrecorded  preparation  for  an  impetus  that  has  given  enduring 
reputation  to  its  standard-bearer.  The  moulding  of  dynamics 
therefore  is  not  the  merit  of  its  master-builders  alone;  we  must 
not  ignore  those  who  had  an  inconspicuous  share  in  establishing 
and  in  perpetuating  its  governing  traditions. 

Then  secondly  it  may  prove  misleading  to  speak  exclusively 


Introductory  Summary  15 

of  changes  and  innovations,  though  some  temporary  aim  compels 
that.  So  we  should  return  to  the  thesis  of  our  opening  para- 
graphs and  allow  them  a  corrective  weight:  That  the  large 
body  of  principles  acquired  early  for  dynamics  and  since  un- 
questioned has  steadied  its  course.  It  has  been  capable  of 
assimilating  the  material  that  we  have  chosen  to  mention  more 
explicitly  without  sacrifice  of  comparative  power  to  treat  for 
example  the  mechanics  of  solids  and  fluids.  The  considerations 
derived  within  that  older  territory  must  hold  their  place  in  what 
now  follows. 

11.  It  will  be  helpful  in  the  direction  of  forestalling  verbal 
quibbles  and  of  clearing  the  ground  otherwise  if  we  enter  next 
upon  an  explanation  of  the  usage  that  we  shall  adopt  for  a  few 
convenient  terms;  and  also  proceed  to  indicate  the  general 
attitude  chosen  in  which  to  approach  mathematical  physics,  of 
which  dynamics  forms  one  part.  It  may  be  well  to  premise 
once  for  all  that  no  such  personal  choice  covers  a  mistaken  en- 
deavor to  close  a  question  that  is  regarded  reasonably  as  open, 
and  to  silence  dissenting  opinion.  But  there  is  often  a  practical 
necessity  for  taking  a  definite  position,  where  adherence  to  one 
view  colors  exposition;  and  thus  it  should  be  candidly  an- 
nounced, although  the  occasion  is  not  appropriate  for  extended 
argument. 

In  accordance  with  the  unavoidable  compulsion  to  take  up 
piecemeal  the  phenomena  and  the  processes  given  by  observation 
and  experiment  in  the  physical  world,  any  particular  problem 
of  dynamics  is  obliged  to  concern  itself  with  a  solution  obtained 
under  recognized  limitations.  These  exhibit  themselves  on  one 
side  in  setting  a  boundary  to  the  region  within  which  the  course 
of  events  shall  be  investigated.  If  we  distinguish  within  such  a 
boundary  a  part  enclosed  that  is  ponderable  and  a  part  that  is 
imponderable,  we  shall  apply  those  terms  on  a  plain  etymological 
basis;  so  that  the  ponderable  contents  have  weight  as  evidenced 


16  Fundamental  Equations  of  Dynamics 

by  the  balance  and  are  subject  to  gravitation,  while  the  im- 
ponderable contents  are  not  thus  detectable.  We  shall  speak 
of  the  former  also  as  masses  or  as  bodies.  The  latter  if  not 
alluded  to  as  free  space  are  called  the  ether,  or  the  medium, 
meaning  the  medium  for  the  transmission  of  light  and  other 
electromagnetic  action.  It  is  assumed  that  the  ether-medium 
has  not  mass  in  the  sense  just  specified;  but  this  does  not  deny 
to  it  the  more  inclusive  quality  of  inertia  in  certain  connections. 
A  distinction  need  not  be  always  upheld  between  mechanics  and 
dynamics ;  but  where  this  is  done  the  second  name  has  the  broader 
scope,  in  that  it  may  bring  both  masses  and  medium  under 
consideration,  which  comprise  then  a  dynamical  system  rather 
than  a  mechanical  one.  By  contrast  the  older  branch,  me- 
chanics, attempts  only  to  deal  with  masses  grouped  into  one 
body,  or  into  a  system  of  bodies.  We  shall  conceive  a  body  to 
fill  its  volume  continuously  and  therefore  to  be  adapted  in  so 
far  to  expressing  by  means  of  an  integral  its  total,  either  of  mass 
or  of  any  quantity  that  is  a  function  of  the  mass-distribution. 
The  conception  behind  the  phrase  system  of  bodies  is  somewhat 
flexible;  it  may  denote  a  discrete  arrangement  of  bodies,  whose 
mass  and  the  like  are  then  given  as  a  sum  of  a  finite  number  of 
terms,  of  which  usage  the  astronomical  view  of  our  sun  and  its 
planets  grouped  as  bodies  in  the  solar  system  affords  a  typical 
instance;  but  it  is  applied  also  to  a  closely  articulated  assemblage 
of  bodies  like  a  machine,  under  suppositions  that  might  or  might 
not  naturally  justify  integration.  The  opposition  between  body 
and  system  of  bodies  is  retained  and  does  some  service  though 
it  is  not  tenable  under  stricter  scrutiny,  and  cannot  be  radical  so 
long  as  physical  theory  actually  analyzes  all  accessible  bodies 
into  fine-grained  systems  for  the  purposes  of  molecular  and 
atomic  dynamics.  On  the  other  hand  the  contrast  between 
systems  of  bodies  and  dynamical  systems  loses  somewhat  in 
significance  where  the  interspaces  are  assumed  to  be  void  and 


Introductory  Summary  17 

the  ether-medium  is  ignored;  an  abstraction  common  every- 
where but  in  electro-magnetism;  and  the  epithet,  dynamical, 
then  points  only  towards  inclusion  of  all  transformations  of 
energy  that  remains  associated  with  masses. 

12.  The  tangled  complexity  in  phenomena  as  they  occur 
however  compels  our  official  accounts  of  them  to  be  given  piece- 
meal in  other  respects  than  by  isolation  of  the  region  that  lies 
within  an  assigned  boundary.  What  is  further  to  be  done  may 
be  denominated  variously;  but  it  runs  toward  idealizing  condi- 
tions, both  by  selecting  certain  elements  as  most  important  for 
study  of  their  quantitative  consequences  and  by  a  restatement 
of  these  that  consciously  relaxes  somewhat  precision  of  corre- 
spondence with  the  facts.  It  is  evident  how  the  two  sources  of 
distortion  are  likely  to  conspire  in  simplifying  the  mathematics; 
since  neglecting  weaker  influences  puts  aside  their  smaller  effects 
as  mere  modifying  terms  of  a  main  result.  To  prune  difficulties 
by  this  procedure  as  a  preliminary  to  formulation  and  discus- 
sion is  in  some  sort  a  contrivance  of  approximation,  conceding 
the  lack  of  desirable  full  power  in  our  mathematical  machinery. 
That  several  determining  reasons  blend  in  it  can  perhaps  be 
recognized,  though  that  is  a  subtle  question  upon  which  we  shall 
not  touch;  but  what  has  practical  weight  is  to  separate  two  uses 
of  approximation,  if  such  omission  be  accepted  as  one  of  them, 
at  the  same  time  granting  that  both  are  drawn  upon  partly 
because  mathematics  limps. ^ 

To  put  the  case  briefly,  sometimes  we  lay  down  a  rule  strictly 
but  approximate  to  the  results  of  it;  which  is  a  purely  mathe- 
matical operation,  utilizing  for  example  a  convergent  series  as 
we  do  when  calculating  the  correction  for  amplitude  in  the  period 
of  a  weight  pendulum.  Or  again  the  assumed  rule  itself  is  known 
to  be  approximate,  as  is  the  fact  when  we  call  the  pendulum 
rigid  and  the  local  weight-field  uniform  and  constant.  A  further 
>  See  Note  5. 


18  Fundamental  Equations  of  Dynamics 

distinction  is  that  the  first  type  relates  to  obstacles  which  may 
be  overcome  entirely  by  device,  as  in  reducing  finally  some 
obstinate  integral,  but  which  lie  ofif  the  track  of  advances  in 
physics.  In  the  instance  just  quoted  the  correction  for  ampli- 
tude will  remain  untouched,  because  an  angle  and  its  sine  will 
never  be  equal.  But  with*  approximations  of  the  second  or 
physical  type  it  is  otherwise;  we  cannot  make  a  body  more  accu- 
rately rigid  by  taking  thought,  nor  can  we  bestow  upon  the 
field-vector  (g)  any  quality  of  constancy  that  it  lacks;  so  they 
progress  by  changing  their  rule.  If  provisional  and  marking 
imperfect  knowledge  while  we  await  amendments  of  magnitude 
not  yet  ascertained,  they  move  toward  refinement  of  precision 
parallel  to  the  advancing  front  of  experimental  research,  as  the 
law  of  Van  der  Waals  about  gases  is  seen  to  improve  upon  that 
of  Boyle.  Yet  no  supreme  obligation  is  felt  to  make  such  changes 
everywhere;  permanent  and  voluntary  renunciations  of  achiev- 
able accuracy  are  frequent,  too;  we  shall  probably  continue  in 
many  connections  to  discuss  rigid  solids  and  ideal  fluids,  not- 
withstanding the  volume  of  fruitful  investigations  in  elasticity, 
in  viscosity  and  elsewhere,  whose  data  are  now  at  our  disposal. 

13.  All  these  points  are  self-evident  at  first  contact,  and  yet 
it  is  advisable  to  name  them,  in  order  to  put  aside  what  is  inci- 
dental and  focus  attention  upon  the  intrinsic  structure  of  our 
equations,  which  leaves  them  inevitably  approximate  as  an 
accepted  limitation  due  to  idealized  or  simplified  statement. 
Clothing  this  thought  in  a  figure,  let  us  say  that  the  principles  of 
physics  crystallize  from  the  data  of  discovery  into  the  concepts 
that  have  been  shaped  by  invention  to  express  them,  but  not 
without  revealing  traces  of  constraint  and  distortion  that  are 
not  subdued  and  made  quite  to  vanish  under  repeated  attempts 
at  adjustment.  Historical  inquiry  has  brought  to  light  some 
remarkable  interdependences  here,  and  furnished  a  list  of  ex- 
amples how  discovery  has  stimulated  the  invention  of  concepts 


Introductory  Summary  19 

to  match,  and  how  on  the  other  hand  a  stroke  of  inspiration  in 
devising  a  well-adapted  concept  has  smoothed  the  path  to  dis- 
covery of  principle.  Nevertheless  the  intimate  psychology  of 
such  reciprocity  is  one  of  those  deep  secrets  that  have  been 
securely  guarded,  and  it  need  not  concern  us;  we  reach  the  kernel 
of  the  matter  for  the  present  connection  when  we  insist  upon  the 
framework  of  dynamics  as  built  of  invented  concepts  and  add 
one  or  two  corollaries  of  that  central  idea. 

In  the  first  place,  in  order  to  proceed  bj'^  mathematical  reason- 
ing from  specified  assumptions,  the  margin  of  ambiguity  in  the 
terms  that  are  used  must  be  cut  down  as  much  as  is  feasible. 
A  controversy  about  Newton's  third  law;  whether  or  not  it 
applies  to  a  source  of  light,  could  be  settled  easily  under  our 
agreement  that  the  ether-medium  is  not  a  body  {corpus).  And 
the  emancipation  from  corroborative  tests  in  the  free  realm  of 
concepts  is  some  compensation  for  the  trouble  of  defining.  It 
has  been  laborious  to  disentangle  the  mean  solar  second  as  a 
uniform  standard  of  time;  but  the  fluxion-time  (t)  of  Newton  in 
its  quality  of  independent  variable  must  be  equicrescent.  So  in 
the  concept  of  unaccelerated  translation  there  is  no  place  for 
differences  of  velocity  anywhere  or  at  any  time;  and  values 
specified  to  be  simultaneous  cannot  be  affected  by  uncertain 
deviations  from  that  assumption;  and  for  the  conceptual  iso- 
tropic solid  under  Hooke's  Law  the  stress-strain  relation  is 
rigorouslj'^  linear.  Likewise,  if  according  to  the  tenets  of  rela- 
tivity the  light-speed  in  free  space  and  relative  to  the  source  is 
always  the  same,  we  go  on  unflinching  to  work  out  the  conse- 
quences; and  any  such  assumption  with  its  demonstrable  deduc- 
tions will  be  entertained  with  candor,  so  long  as  its  contacts  with 
observed  facts  given  by  correct  mathematics  do  not  fail  either 
as  plausible  physics.  However,  from  the  side  of  these  perpetual 
tests  there  is  sleepless  critical  judgment  upon  all  our  mental 
devices,  to  continue,  to  revise  or  to  reject  them.     In  other  respects 


20  Fundamental  Equations  of  Dynamics 

the  schemes  may  be  plastic  to  shift  the  point  in  precision  at  which 
they  halt,  and  we  are  reasonably  tolerant  also  of  conventional 
fictions. 

This  brings  to  a  close  the  short  preface  of  such  verbal  comment 
as  may  provide  a  setting  in  which  to  frame  the  equations  that 
follow,  and  at  the  same  time  assist  in  some  respects  to  receive 
more  appreciatively  their  meaning  by  bringing  to  view  what 
underlies  them. 


CHAPTER   II 
The  Fundamental  Equations 

14.  Any  standard  exposition  of  dynamics,  though  it  may  not 
attempt  a  comprehensive  and  most  general  treatment  of  the 
methods  and  principles,  will  introduce  into  its  resources  for 
carrying  on  the  discussion  the  six  quantities :  Force,  Momentum, 
Kinetic  Energy,  Power,  Force-moment  and  Moment  of  Momen- 
tum. The  terms  in  detail  that  are  required  for  the  specification 
of  these,  and  a  certain  group  of  propositions  into  which  they 
enter,  are  so  fundamental  that  they  become  practically  in- 
dispensable in  establishing  the  necessary  developments.  The 
units  that  their  function  as  measured  quantities  demands  are 
supplied  according  to  the  centimeter-gram-second  system  with  so 
nearly  universal  adoption  that  we  can  regard  it  as  having  dis- 
placed all  competitors,  everywhere  except  in  some  technical 
applications  where  special  needs  prevail;  so  that  we  shall  con- 
sider no  alternative  plan  of  measurement. 

Since  the  six  quantities  named  are  not  independent  of  each 
other,  but  are  connected  by  a  number  of  cross-relations  that  we 
can  assume  to  be  famiHar  in  their  elementary  announcement,  it 
is  clear  that  the  way  hes  open  to  select  for  a  starting-point  a 
certain  set  as  primary,  the  others  then  falling  into  their  own 
place  as  derived  or  even  auxiliary  quantities.  It  is  also  plain, 
as  a  mere  matter  of  logical  arrangement,  that  any  particular 
selection  of  a  primary  set  will  not  be  unique,  with  a  monopoly 
of  that  title  to  be  put  first;  and  this  leaves  the  exercise  of  prefer- 
ence to  be  governed  ultimately  by  reasons  drawn  from  the 
subject-matter.  Not  only  is  it  possible  to  make  beginnings 
from  more  points  than  one  in  presenting  the  six  quantities  on  a 

21 


22  Fundamental  Eqiiations  of  Dynamics 

definite  basis,  and  in  exhibiting  the  Hnks  among  them,  but  it  is  the 
truth  that  beginnings  have  been  made  differently  and  defended 
vigorously.  We  have  already  alluded  to  one  such  period  of 
polemic  through  which  dynamics  has  passed.  It  is  a  necessity 
however  to  choose  a  procedure  by  some  one  line  of  advance; 
but  let  it  be  understood  that  we  do  this  with  no  excessive  claim 
for  its  preponderant  advantage  or  convenience,  and  explicitly 
without  prejudice  to  the  validity  of  some  other  sequence  that 
may  be  preferred. 

15.  In  the  light  ol  this  last  remark  we  shall  make  our  start 
by  picking  out  for  first  mention  a  group  of  three  quantities: 
Momentum,  Kinetic  Energy  and  Moment  of  Momentum.  With- 
out anticipating  a  more  specific  analysis  of  them,  it  is  evident  on 
the  surface  that  they  all  apply  in  designating  an  instantaneous 
state  depending  on  velocities,  and  that  momentum  is  the  core 
of  the  three;  entering  as  free  vector,  as  localized  vector,  and  as 
factor  in  a  scalar  product.  And  further  it  can  be  noted  at  once, 
without  presuming  more  than  a  first  acquaintance  with  me- 
chanics, that  the  remaining  three  quantities  constituting  a  second 
group  can  be  described  in  symmetrical  relation  to  the  first  three 
as  their  time-rates.  Then  force  is  made  central;  and  it  in  turn 
appears  as  a  free  vector,  as  a  localized  vector,  and  as  a  factor  in  a 
scalar  product.  We  take  the  first  step  accordingly  by  laying 
down  for  application  to  any  body  or  to  any  system  of  bodies  the 
three  defining  equations : 

Total  momentum  =  S  /mVdm  =  Q ;  (I) 

Total  kinetic  energy  =  S  fjn^(y-vdm)  =  E;  (II) 

Total  moment  of  momentum  =  S  /m(r  x  vdm)  =  H.  (Ill) 
These  indicate  in  each  case,  with  notation  that,  is  so  nearly 
standard  as  to  carry  its  own  explanation,  the  result  of  a  mass- 
summation  extended  to  contributions  from  all  the  mass  included 
in  the  system  at  the  epoch,  under  the  terms  of  some  agreement 


The  Fundamental  Equations  23 

covering  the  particular  matter  in  hand,  and  isolating  in  thought 
temporarily,  for  purposes  of  study  and  discussion,  the  phenomena 
in  a  limited  region.  In  conformity  with  a  previous  explanation 
in  section  11  any  assumed  continuous  distributions  of  mass  are 
included  under  the  integrals,  whose  further  summation  indicated 
by  (2)  may  be  necessary  when  a  system  of  bodies,  discrete  or 
contiguous,  is  to  be  considered.  It  deserves  to  be  emphasized 
perhaps  that  these  are  defining  equalities  merely;  so  that  (Q) 
and  (H)  and  (E)  only  denote  aggregate  values  associated  with 
the  system  at  the  epoch,  and  so  to  speak  observable  in  it;  neither 
side  of  the  equalities  conveys  any  implication  about  external 
sources,  or  causes  by  whose  action  these  aggregates  may  have 
originated,  or  which  may  be  operative  at  that  epoch  to  bring 
about  changes  affecting  them. 

16.  Because  the  variables  (r)  and  (v)  occur  in  the  quantities 
with  which  we  are  now  dealing,  if  for  no  deeper  reason,  it  is 
implied  that  a  definite  system  of  reference  has  been  fixed  upon 
as  an  essential  preliminary  to  actual  attachment  of  values  to 
momentum,  kinetic  energy  and  moment  of  momentum.  For 
the  ordinary  routine  which  is  likely  to  involve  recasting  vector 
statements  into  semi-cartesian  equivalents,  or  the  inverse  opera- 
tion of  arriving  at  the  former  by  means  of  the  latter,  the  requisite 
elements  for  the  reference  are  obtained  by  selecting  an  origin 
from  which  to  measure  distances  and  axes  for  orienting  directions. 
Unless  special  exception  be  explicitly  noted  we  shall  follow  the 
prevalent  usage  of  taking  axes  of  reference  that  are  orthogonal 
and  in  the  cycle  of  the  right-handed  screw;  and  shall  for  con- 
venience conduct  the  main  discussion  on  this  permanent  back- 
ground, reserving  any  substitution  of  equivalents  for  occasions 
where  that  has  some  peculiar  fitness.  The  reference-frame  that 
has  been  agreed  upon,  it  must  not  be  forgotten,  is  in  the  essence 
of  it  conceptually  fixed  while  the  agreement  to  use  it  continues  in 
force,  because  it  has  been  singled  out  as  the  unique  standard  in 
3 


24  Fundamental  Equations  of  Dynamics 

relation  to  which  we  specify  or  trace  what  can  be  called  the 
configurations  (r)  and  the  motions  (v). 

As  an  antecedent  condition  of  algebraic  evaluation  for  our 
three  fundamental  quantities  in  a  given  system  at  any  epoch, 
the  choice  of  some  reference-frame  then  is  necessary;  but  it  is 
likewise  evident  that  any  one  choice  that  may  be  made  is  equally 
sufficient  in  respect  to  removing  mathematical  indeterminateness. 
And  consequently  it  will  be  found  true  that  much  can  be  done 
in  advancing  a  satisfactory  exposition  of  dynamical  principles  to 
the  point  where  we  stand  at  the  threshold  of  calculations  that 
rest  on  a  basis  of  observed  phenomena,  without  going  beyond  the 
potential  assumption  of  that  reference-frame  that  must  be  faced 
finally,  in  order  to  complete  the  necessary  and  sufficient  condi- 
tion for  the  definiteness  of  the  physical  specifications.  In  other 
words,  a  considerable  proportion  of  the  usual  developments  in 
dynamics  can  be  provided  ready-made  to  this  extent,  and  yet 
fitting  the  measure  of  any  reference-frame  that  is  particularly 
indicated  as  appropriate  by  a  physical  combination  or  by  a  line 
of  argument.^ 

These  considerations  are  adapted  to  bring  to  the  front  also 
the  idea  that  quantities  like  the  three  with  which  we  are  con- 
cerned at  tnis  moment  can  be  evaluated  for  two  or  more  different 
reference-frames,  perhaps  with  the  object  of  reviewing  their 
comparative  merit,  especially  in  being  adjusted  to  the  preferences 
of  consistent  physical  views  (see  section  7).  It  follows  naturally 
therefore  that  provision  must  be  made  quantitatively  for  trans- 
fers of  base  from  one  reference-frame  to  another,  either  in  progress 
toward  ultimate  reference,  as  in  abandoning  a  frame  fixed 
relatively  to  the  rotating  earth,  or  as  a  device  of  ingenuity  in 
order  to  reach  certain  ends  simply.  The  material  of  Chapter  iii 
in  large  part  bears  upon  questions  of  that  nature. 

17.  The  range  of  the  mass-summations  that  are  stipulated  in 

1  See  Note  6. 


The  Fundamental  Eqiiations  25 

the  expressions  with  which  we  are  dealing  can  vary  with  time 
for  several  reasons  that  can  be  operative  separately  or  con- 
currently. It  is  compatible  with  many  conditions  about  bound- 
ary-surface that  material  may  be  added  or  lost,  as  is  the  case 
when  gas  is  pumped  into  a  tank  or  out  of  it,  or  when  unit  volume 
of  an  elastic  solid  gains  or  loses  by  compression  or  extension. 
Or  it  may  fit  the  circumstances  best  to  mark  off  a  boundary  that 
changes  with  time,  as  when  we  take  up  mechanical  problems 
Uke  those  of  a  growing  raindrop  or  a  falling  avalanche.  The 
values  of  (Q,  E,  H)  are  accommodated  to  any  complication  of 
such  conditions,  with  the  single  caution  that  the  total  mass  shall 
then  be  delimited  as  an  instantaneous  state  at  the  epoch. 

We  go  on  to  assume,  however,  in  connection  with  any  transfers 
of  reference  that  we  are  called  upon  to  execute,  that  mass  remains 
unaffected  thereby  in  its  differential  elements  and  in  its  total, 
being  guided  by  the  absence  of  experimental  evidence  that  mass, 
in  our  adopted  use  of  the  word,  needs  to  be  made  dependent  upon 
position  or  velocity.  Assembling  these  suppositions,  we  see  that 
mass  will  play  its  part  in  the  equations  as  a  pure  scalar  and 
positive  constant,  except  as  accretions  or  losses  of  recognizable 
portions  may  be  a  feature  of  the  treatment.  And  consequently 
equation  (II)  can  be  made  algebraic  at  once,  since  the  vector 
factors  are  codirectional,  and  be  given  the  form 

E  =  2  /^(IvMm),  (1) 

although  the  original  model  should  be  preserved  besides,  as  a 
point  of  departure  for  parallelisms  that  will  show  themselves  later. 
18.  Return  now  to  examine  the  two  remaining  equations,  in 
order  to  extract  some  additional  particulars  of  their  meaning. 
In  the  first  the  total  momentum  appears  as  a  vector  sum,  so 
built  up  that  its  constituents  are  usually  described  as  free  vectors. 
This  term  is  seen  to  justify  combining  the  dispersed  elements  to 
one  resultant,  on  reflecting  that  the  predicated  freedom  of  such 


26  Fundamental  Equations  of  Dynamics 

vectors  lies  wholly  in  the  non-effect  of  mere  shift  to  another  base- 
point;  and  that  this  renders  legitimate  the  indefinite  repetition  of 
the  parallelogram  construction  for  intersecting  vectors  until  all 
the  differential  elements  have  been  absorbed,  into  the  total  aggre- 
gate. But  this  incidental  and  as  it  were  graphical  convenience 
must  not  lead  us  to  neglect  the  fact  that  we  are  nevertheless 
retaining  the  idea  of  momentum  as  a  distributed  vector,  and  con- 
tinuing to  associate  each  element  of  it  locally  with  some  element 
of  mass.  However  formed  its  total  belongs  to  the  system  as  a 
whole;  and  it  can  be  localized,  as  it  sometimes  is  at  the  center 
of  mass  of  the  system,  only  by  virtue  of  a  convention  or  an 
equivalence.^ 

We  can  call  the  total  momentum  a  free  vector,  of  course;  but 
its  freedom  does  not  quite  consist  in  an  indifference  about  its 
base-point;  more  nearly  it  expresses  the  inherent  contradiction 
there  would  be  in  localizing  anywhere  what  in  fact  is  still  con- 
ceived to  pervade  the  mass  of  the  system.  At  several  points  we 
5hall  discover  how  the  service  of  vectors  in  physics  makes  desir- 
able some  addition  to  the  formal  mathematical  handling  of  them. 
It  will  not  be  overlooked,  finally,  how  the  above  analysis  of  com- 
position enlarges  upon  the  addition  qf  parallel  forces  to  constitute 
.a  total,  through  the  similar  properties  of  an  algebraic  and  a  geo- 
metric sum;  the  latter  reduces  to  its  resultant  by  complete 
cancellation  in  a  plane  perpendicular  to  the  resultant. 

19.  In  the  third  equation  each  local  element  of  momentum 
has  the  attribute  of  a  localized  vector  through  definite  assign- 
ment to  the  extremity  of  its  radius-vector.  It  is  not  apparent 
that  the  vector  product  in  which  it  is  a  factor  is  thereby  deter- 
mined to  be  unequivocally  localized;  but  here  again  physical 
considerations  enter  that  are  extraneous  to  the  mathematics;  the 
practice  tacitly  followed  localizes  the  several  elements  of  moment 
of  momentum,  not  at  the  differential  masses  to  which  they  in 

1  See  Note  7. 


The  Fundamental  Equations  27 

one  sense  belong,  but  at  the  origin  in  acknowledgment  of  their 
intimate  connection  with  rotations  about  axes  there,  and  of  the 
origin's  importance  in  determining  the  lever-arms  when  the 
mass-arrangement  is  given.  Each  differential  moment  of 
momentum  thus  located  being  perpendicular  to  the  plane  of  its 
(r)  and  (v)  of  that  epoch,  is  evidently  also  normal  to  the  plane 
containing  consecutive  positions  of  the  radius-vector;  that  is, 
(dH)  is  colinear  with  (dy),  if  the  latter  denotes  the  resultant 
element  of  angle- vector  that  (r)  is  then  describing;  and  on  this 
we  can  found  a  transformation  that  is  worth  noting.  If  (ds)  is 
the  element  of  path  for  (dm), 

dr  =  ^  (r  X  ds);  ^  =  ^^  (r  X  v);        dH  =  Y(rMm);    (2) 

and  the  last  equation  reproduces  differentially  the  type  of  an 
elementary  and  partial  relation  among  moment  of  momentum, 
moment  of  inertia  and  angular  velocity  for  a  rigid  solid.  Only 
(y)  is  here  individually  determined  in  magnitude  and  in  direction 
for  each  (dm);  no  common  angular  velocity  and  collective 
moment  of  inertia  are  assigned,  as  they  are  in  the  case  of  a 
rigid  solid,  but  with  disturbance  in  general  of  the  colinearity 
shown  by  (dH)  and  (y)  into  h  divergence  of  the  resultant  vectors 
for  angular  velocity  and  moment  of  momentum. 

20.  The  three  equations  of  section  15  are  simplified  remarkably 
whenever  the  condition  prevails  that  the  velocity  (v)  has  a 
common  value  throughout  the  system  that  is  in  question.  This 
state  of  affairs  is  designated  as  translation  of  the  system;  it  may 
persist  during  a  finite  interval  of  time,  or  it  may  appear  only 
instantaneously,  and  in  either  case  naturally  it  entails  a  corre- 
sponding quality  in  the  simplifications.  When  the  condition  of 
translation  persists  the  common  velocity  (v)  need  not  be  con- 
stant; but  the  simultaneous  velocities  everywhere  must  be 
equal.  The  resulting  forms  applying  to  translation  are  then 
seen  to  be  for  a  total  mass  (m). 


28  Fundamental  Equations  of  Dynamics 

Q  =  vS  /^dm  =  mv;  (3) 

E  =  §(vv)S/^dm  =  imv2;  (4) 

H  =  (S  fjjdm)  X  V  =  f  X  mv.  (5) 

The  last  equation  introduces  the  familiar  mean  vector  (f)  which 
locates  the  center  of  mass  of  the  system  through  the  mass- 
average  of  the  individual  radius-vectors  (r)  according  to*  the 
defining  equation 

mf  ^  S  /„,rdm.  (6) 

The  last  group  of  equations  contains  the  suggestion  from  which 
has  been  worked  out  a  notion  that  has  had  some  vogue  and 
convenience  in  dynamics:  that  of  an  equivalent  or  representative 
particle  to  which  are  attributed  negligible  dimensions  but  also 
the  total  mass,  momentum  and  kinetic  energy  of  the  system. 
Equations  (3,  4,  5)  show  that  such  a  fictitious  particle  at  the 
position  of  the  center  of  mass  of  the  system  would  replace  the 
latter  in  respect  of  (Q,  E,  H)  while  translation -continues.  And 
since  it  is  their  ratio  to  other  lengths  that  settles  whether 
dimensions  are  physically  negligible,  the  absurdity  that  there 
would  be  in  concentrating  momentum  and  energy  into  a  mathe- 
matical point  is  sensibly  mitigated. 

21.  Even  when  the  condition  is  not  met  that  simultaneous 
velocities  shall  be  equal  everywhere,  a  constituent  translation 
can  be  carved  out  artificially  from  the  actual  totals  (Q,  E,  H) 
at  the  epoch  and  for  the  system.  Let  every  local  velocity  (v) 
be  split  into  two  components  in  conformity  with  the  relation 

V  =  c  +  u,  (7) 

in  which  (c)  is  assigned  at  will,  but  taken  everywhere  equal,  and 
(u)  denotes  whatever  remains  of  (v).  Then  substitution  in  the 
fundamental  equations  of  section  15  will  segregate  the  totals 
into  a  part  that  corresponds  to  translation  and  a  supplement. 
Among  the   indefinite  number  of  possibilities,   we  select   one 


The  Fundamental  Equations  29 

particularly  fruitful  plan  for  illustration.  Let  (v)  be  the  mass- 
average  of  velocities  determined  by  the  condition 

mv  =  2  /^vdm.  (8) 

Then  if 

V  =  V  +  u,         2  /mUdm  =  0  necessarily.  (9) 

But  we  have  also,  in  consequence  of  equation  (9), 

E  =  is  /„,(v  +  u)  •  (v  +  u)dm  =  imv2  +  |S  /^uMm.    (10) 

And  further, 

H  =  2  /Jr  X  (V  +  u)dm]  =  (2  /^rdm)  x  v  +  2  /^(r  x  udm).  (11) 

In  order  to  reduce  the  last  term  place  r  =  ir  +  r',  so  that  (r') 
like  (u)  is  departure  of  the  local  value  from  the  mean.  Then 
finally 

H  =  (f  X  vm)  +  2  /^(r'  x  udm),  (12) 

in  which  the  segregation  according  to  mean  values  and  de- 
partures from  them  is  complete. 

Taking  equation  (8)  in  conjunction  with  the  first  terms  on  the 
right-hand  of  equations  (10)  and  (12).  the  idea  of  a  particle  at  the 
position  of  the  center  of  mass  reappears,  having  the  total  mass  (m) , 
the  total  momentum  (mv  =  Q),  and  the  kinetic  energy  (^mv^). 
But  whereas  equations  (3,  4,  5)  covered  the  data  completely,  this 
contrived  and  partial  translation  with  the  mean  velocity  (v) 
leaves  residual  amounts  of  kinetic  energy  and  moment  of  momen- 
tum; and  these  are  due  to  departures  from  the  mean  values  of  (r) 
and  (v),  as  the  last  terms  in  equations  (10)  and  (12)  indicate 
plainly,  which  items,  as  is  also  evident,  have  no  resultant  influ- 
ence on  the  momentum.  It  is  clear  that  this  plan  of  partition  is 
adapted  to  accurate  use;  but  it  proves  to  have  some  advantages 
too  as  the  basis  of  an  approximation,  where  the  residual  terms 
are  in  small  ratio  to  the  translation-quantities  and  can  be  ne- 
glected in  comparison  with  the  latter.  The  so-called  simple 
'pendulum  affords  one  instance. 


30  Fundamental  Equations  of  Dynamics 

22.  The  recognition  of  elements  of  momentum  as  localized 
vectors  brings  in  an  additional  detail  of  their  physical  specifica- 
tion; so  this  alone  could  be  alleged  as  one  valid  reason  for  con- 
ceding to  moment  of  momentum  its  place  in  the  general  founda- 
tion of  dynamics.  But  we  are  now  in  a  position  to  realize  another 
advantage  of  which  that  third  equation  gives  us  control.  Mean 
values  have  admitted  elements  of  strength  in  smoothing  out 
accidental  or  systematic  differences  in  a  series  of  data,  and  in 
enabling  us  to  convert  an  integral  into  a  product  of  finite  factors. 
Yet  this  acceptable  aid  may  be  offset  in  part  by  such  elimination 
on  the  whole  of  departures  from  the  mean  as  is  shown  in 

S/^r'dm  =  0;         2 /„udm  =  0. 

Now  first  inspection  of  equation  (12)  shows  how  it  serves  to 
retrieve  by  means  of  the  vector  products  the  divergencies  that 
would  be  lost  from  sight  in  the  mean  values,  and  thus  to  piece 
out  the  support  in  that  direction  which  equation  (10)  accom- 
plishes through  its  scalar  products,  wherever  we  have  an  interest 
to  gauge  effects  of  divergence  that  are  cumulative  and  not 
self-canceUing, 

23.  Before  passing  on  to  another  topic  it  is  worth  taking  occa- 
sion to  remark  that  the  values  for  the  totals  of  momentum, 
kinetic  energy  and  moment  of  momentum  can  be  adjusted 
without  difficulty  to  expression  as  summations  extended  over  a 
volume;  for  in  terms  of  the  local  density  (5)  and  element  of 
volume  (dV)  the  mass-element  there  is  expressible  by 

dm  =  5dV. 

This  density  will  be  rated  always  a  pure  scalar  on  ac^count  of 
its  correlation  with  mass,  and  both  density  and  volume  are  best 
standardized  in  dynamics  as  positive  factors  in  the  positive 
product  that  is  mass,  though  it  is  not  advisable  to  brush  aside 
too  lightly  the  combinations  that  the  character  of  volume  as  a 


The  Fundamental  EqvMions  31 

pseudo-scalar  permits.  Since  the  value  of  the  density  is  zero 
throughout  the  volume  that  is  left  unoccupied  by  the  supposed 
distribution  of  mass,  the  inclusion  of  such  portions  into  a  summa- 
tion throughout  the  whole  region  within  the  assumed  boundary 
is  without  influence  upon  the  result  and  can  be  indicated  formally 
without  error.  To  declare  a  density  zero  is  the  equivalent  of 
excluding  a  volume  from  a  mass-summation. 
-  Hence  the  need  of  a  double  notation  (2)  and  ( /m)  will  dis- 
appear, if  the  continuous  volume  can  be  paired  with  a  density 
also  eJBfectively  continuous,  by  any  of  the  plausible  devices  that 
evade  abrupt  changes  at  passage  from  volume  with  which  mass 
is  associated  to  volume  from  which  mass  is  said  to  be  absent.^ 
With  these  words  of  explanation  the  alternative  forms  that  follow 
are  interpretable  at  once: 

Q  =  Ao.v(5dV);  (13) 

E  =  !/,„,  v.v(5dV);  (14) 

H  =  /vo,.(rxv(5dV)).  (15) 

Let  us  add  for  its  bearing  upon  the  lines  of  treatment  when 
mass  is  variable  that  then  both  (5)  and  (dV)  are  susceptible  of 
change.  And  also  recall  how  there  will  always  be  that  out- 
standing question  about  mean  values  in  comparison  with  diver- 
gence from  them,  of  which  we  spoke  above,  whenever  we  face  the 
contradictory  demands  of  mathematical  continuity  and  of  open 
molecular  structure,  in  order  to  reconcile  them  adequately — for 
instance,  in  the  concept  of  a  homogeneous  body  with  a  value  of 
density  that  is  common  to  all  its  parts. 

24.  We  shall  next  approach  the  remaining  group  of  funda- 
mental quantities  that  we  have  enumerated  already  as  three: 
Force,  Power,  and  Force-moment.  The  first  object  must  be  to 
set  forth  in  satisfying  clearness  and  completeness  their  relations 

1  See  Note  8. 


32  Fundamental  Equations  of  Dynamics 

to  the  previous  group  of  three,  in  order  to  proceed  then  securely 
with  reading  the  lesson  how  the  interlinked  and  consolidated 
set  of  six  quantities  provides  all  requisite  solid  and  efficacious 
support,  both  for  the  current  general  reasoning  of  dynamics 
and  for  its  specialized  lines  of  employment. 

We  began  to  follow  the  track  over  ground  that  has  become 
well-trodden  since  Newton's  day,  when  we  laid  down  a  meaning 
for  the  phrase  total  momentum  of  a  system  of  bodies  and  the  symbol 
(Q)  representing  it  which  in  effect  only  renames  the  intention  of 
the  historic  words  "  Quantity  of  motion."  We  also  continue  the 
tradition  that  has  been  perpetuated  ever  since  Newton's  second 
law  launched  its  beginnings  for  approval,  by  fixing  attention  in 
its  turn  upon  the  rate  of  change  in  the  momentum,  in  its  differ- 
ential elements  and  in  its  total,  and  regard  that  as  delivering  to 
us  the  clews,  that  we  shall  later  follow  up,  to  the  forces  brought 
to  bear  upon  the  system  of  bodies  that  is  under  investigation, 
with  which  forces  we  must  undertake  to  reckon.  The  gist  of 
that  law  has  not  yielded  perceptibly  under  all  the  proposals  to 
improve  upon  it,  though  we  may  be  rewording  it  more  flexibly 
under  widening  appeal  to  experience.  Its  drift  makes  the  claim 
that  changes  in  (Q)  are  not  spontaneous;  that  when  they  are 
identified  to  occur  there  is  reason  to  be  alert  and  detect  why, 
with  gain  for  physical  science  in  prospect  by  success. 

25.  The  first  move  toward  formulation  could  scarcely  be 
simpler;  it  is  to  indicate  the  time-derivative  of  equation  (I)  and 
write 

Q-^[2/^vdml.  (IV) 

Yet  the  mere  mathematics  of  execution  blocks  the  way  with 
distinctions  to  be  made,  unless  we  are  resolved  to  carry  an  over- 
weight of  hampering  generality.  For  it  is  common  knowledge 
that  the  masses  are  often  comprised  in  such  a  summation  on  a 
justified  footing  that  they  are  in  every  respect  independent  of 


The  Fundamental  Equations  33 

time;  and  consequentlj'  it  is  then  legitimate  to  differentiate 
behind  the  signs  of  summation  in  equation  (IV).  But  forms 
alter  as  the  mass  included  is  in  any  way  a  function  of  time; 
they  will  differ  besides  if  onh^  the  total  mass  changes  by  loss  or 
gain  of  elements,  or  only  the  elements  change  leaving  the  total 
constant,  or  if  both  sorts  of  dependence  upon  time  are  permissible 
under  the  scheme  of  treatment.  The  first  supposition  of  complete 
mass-constancy  underlies  the  dynamics  of  rigid  solids  and  is  a 
stock  condition  in  much  dynamics  of  fluids  as  well.  And  because 
it  prevails  most  naturally  to  that  extent,  it  is  perhaps  fair  to 
select  this  mass-constancy  as  standard;  especially  when  depar- 
tures from  it  are  likely  soon  to  be  cut  off  from  the  stream  of 
systematic  development  by  running  into  speciaHzing  restrictions 
and  a  narrowly  applicable  result.^ 

However  opinion  may  stand  on  that  matter,  it  seems  certain 
that  no  aspect  should  be  allowed  to  escape  us  finally  that  belongs 
to  the  full  scope  of  mathematical  possibility  attaching  to  the 
indicated  time-derivative  of  (Q).  Anj^  contribution  to  the 
changes  in  momentum  may  mature  a  suggestion  about  force- 
action  and  gain  physical  meaning.  Therefore  the  tendency  seems 
unfortunate  to  borrow  the  terms  of  Newton's  second  law,  for 
its  professedly  general  statement,  from  the  special  though  widely 
prevalent  case  which  throws  all  the  change  in  momentum  upon 
the  velocity-factor.  To  speak  of  force  as  universally  measured 
b}'  the  product  of  mass  and  acceleration  is  misleading  if  the 
habit  blinds  us  to  the  fuller  scope  of  the  second  law,  and  atrophies 
at  all  our  capacity  to  use  it. 

26.  In  order  that  the  derivative  of  an  expression  may  be 
formed  for  use,  certain  conditions  of  continuity  must  not  be 
violated,  as  we  know;  but  when  a  derivative  is  to  be  made 
representative  of  a  sequence  of  states,  mathematical  physics  has 
available  a  repertory  of  resources  in  constructing  this  requisite 

1  See  Note  9. 


34  Fundamental  Equations  of  Dynamics 

continuity  of  duration  and  distribution.  Examples  are  plentiful 
among  the  classic  methods  of  attack,  how  variously  the  proper 
degree  of  identifiable  quality  is  assigned  to  a  succession  of  states, 
that  links  the  individual  terms  into  a  continuous  series.  Rankine's 
device  for  studying  a  sound-wave  in  air  is  a  travelling  dynamics 
that  keeps  abreast  of  the  propagation;  Euler's  hydrodynamical 
equations  stand  permanently  at  the  same  element  of  volume,  and 
record  for  successive  portions  of  liquid  that  stream  by;  and  many 
processes  where  material  passes  steadily  through  a  machine  are 
most  tractable  in  similar  fashion.  We  shall  not  insist  further 
then  upon  this  point,  except  to  say  that  advanced  stages  of  the 
subject  are  less  apt  to  rely  upon  straightforward  sameness  and 
constancy  in  the  masses  specified  for  summation  under  the  term 
body  or  system  of  bodies.  With  the  reserves  of  that  cautious  pre- 
amble, we  can  afford  to  qualify  the  case  of  mass-constancy  and 
literal  sameness  as  standard  in  a  limited  sense,  and  exploit  some 
of  the  consequences  flowing  from  that  assumption. 

27.  On  the  grounds  now  announced  explicitly  the  indicated 
operation  of  equation  (IV)  yields 

Q  =  2  fjdm  ^  S  /^dR  =  R.  (16) 

As  a  symbol,  therefore,  (dR)  is  defined  to  mean  the  local  resultant 
force  at  each  diflerential  mass  for  which  there  is  evidence  through 
the  local  acceleration;  and  accordingly  (R)  denotes  the  vector 
sum  of  such  elements  of  force  when  the  whole  system  of  bodies 
is  included.  This  total  force  is  in  nature  a  dispersed  aggregate 
like  the  total  momentum,  and  the  line  of  comment  under  that 
heading  applies  here  with  a  few  changes,  which  however  are 
obvious  enough  to  absolve  us  from  repeating  it. 

28.  Before  we  carry  the  discussion  into  farther  detail  it  seems 
best  to  bring  equations  (II)  and  (III)  to  this  same  level  by 
putting  down  their  time-derivatives,  observing  consistently 
there  also  the  imposed  limitation  to  complete  mass-constancy, 


The  Fundamental  Equations  35 

but  remembering  always  that  vve  halted  exactly  on  that  line  and 
postponed  until  due  notice  shall  be  given  the  further  step  in 
restriction  that  will  introduce  a  rigidly  unchanging  arrange- 
ment or  configuration  of  all  the  mass-elements.  Writing  first  the 
general  defining  equation  as  preface, 

dE 

we  can  then  make  the  application  to  the  specialized  conditions 
that  gives 

^=|S/^^(vvdm)  =  2/JvdR).  (17) 

This  indicates  at  each  element  of  mass  a  local  manifestation  of 
power  that  is  measured  by  the  scalar  product  of  the  force-element 
and  velocity — this  scalar  product  being  of  course  not  merely 
formal,  since  (v)  and  (dR)  are  not  in  general  colinear.  It  has 
been  called  also,  perhaps  with  equal  appropriateness,  the  activity 
of  the  force. 

29.  In  this  preliminary  consideration  there  remains  only  the 
time-derivative  of  equation  (III) .  And  we  shall  preserve  a  help- 
ful symmetry  of  statement  by  giving  its  place  here  also  to  the 
general  defining  equation,  and  following  it  as  we  have  done 
previously  with  its  present  special  value.     Then 

M  =  H;  (VI) 

and 

M  =  2  /^-  (r  X  vdm)  =  S  /n,(r  x  dR);  (18) 

the  reduction  of  the  expansion  to  one  of  its  two  terms  being  the 
evident  consequence  of  the  identity  of  (r)  and  (v).  The  last- 
equation  demonstrates  within  the  limits  set  for  it  that  the  time- 
derivative  of  the  total  moment  of  momentum  measures  the  total 
force-moment  of  the  local  elements  of  force  that  are  calculated 


36  Fundamental  Equations  of  Dynamics 

according  to  equation  (16).  As  a  postscript  to  equation  (18) 
repeat  with  the  necessary  modifications  what  was  inserted  in 
section  19,  about  equation  (III),  and  at  the  end  of  section  22. 
The  example  of  a  force-couple  will  come  to  mind  at  once,  where 
the  pair  of  its  forces  is  self-cancelling  from  the  free-vector  aggre- 
gate of  force,  and  it  devolves  upon  the  localized  force- vector  of  a 
moment  to  restore  for  consideration  the  important  effects  of 
couples. 

Observe  also  the  peculiar  prominence  of  the  radius-vector  in 
vector  algebra.  Where  the  cartesian  habit  is  to  bring  both 
moment  of  momentum  and  force-moment  into  direct  and  ex- 
clusive relation  to  a  line  or  axis,  vector  methods  substitute  rela- 
tion to  the  origin,  which  is  a  point.  Upon  examination,  however, 
the  difference  partly  vanishes,  because  the  vector  reference  to  a 
point  is  only  a  superficial  feature.  We  have  explained  in  con- 
nection with  equation  (2),  how  a  resultant  axis  is  tacitly  added. 
The  element  (dM)  is  similarly  a  maximum  or  resultant,  the 
factors  in  (r  x  dR)  being  given,  the  effective  fraction  of  the 
moment  for  other  axes  through  the  origin  being  obtainable  by 
projecting  (dM).^ 

30.  Equations  (16,  17,  18)  bear  on  their  face  and  for  their 
particular  setting  sufficient  reasons  for  interpreting  (Q)  in  terms 
of  those  forces  (dR);  (P)  or  (dE/dt)  in  terms  of  the  activity  of 
those  same  forces  (v-dR);  and  (M)  or  (H)  in  terms  of  their 
force-moments  (r  x  dR).  There  seems  to  be  neither  confusion 
nor  danger  imminent  if  we  extend  the  names  thus  rooted  in 
commonplace  experience  to  the  (at  least  mathematically)  more 
complicated  possibilities  of  equations  (IV),  (V),  (VI).  We  can 
be  bold  to  identify  (Q)  always  as  some  force  (R);  (dE/dt)  as  a 
power  (P);  and  (H)  always  as  a  force-moment  (M);  if  we  have 
made  ourselves  safely  aware  how  terms  in  any  completed  mathe- 
matical expansion  may  remain  non-significant  physically  until 

1  See  Note  10. 


The  Fundamental  Equations  37 

discovery  confirms  them.  We  have  alluded  before  to  the  fact 
that  dynamics  does  not  altogether  shrink  from  a  figurative  tinge 
in  extensions  of  terms  first  assigned  Uterally,  if  essentials  of 
correspondence  are  adequately  preserved.  But  notice  particu- 
larly that  the  verbal  adoptions  proposed  above  cannot  of  them- 
selves assure  the  occurrence  of  the  duplicate  adjustments  among 
equations  (16),  (17),  and  (18).  To  forces  whose  sum  is  (Q)  will 
correspond  activities  that  we  may  denote  by  (v-dQ),  and  mo- 
ments of  type  (r  x  dQ).  But  we  must  not  conclude  in  advance 
that  the  former  group  will  in  their  sum  match  (dE/dt) ;  nor  that 
the  latter  group  will  match  exactly  (H);  though  both  equiva- 
lencies hold  under  the  condition  of  mass-constancy.  And  for 
discrepancies  there  will  be  no  general  corrective  formula;  they 
must  be  newly  weighed  wherever  they  may  appear. 

31.  Let  us  next  turn  back  to  the  ideas  of  translation  and 
equivalent  particle  of  which  we  spoke  in  sections  20  and  21, 
and  continue  them  in  the  light  of  equations  (16,  17,  18).  In  the 
first  place  note  that  the  mean  velocity  (v)  as  previously  specified 
by  equation  (8)  becomes  now  identical  with  the  velocity  of  the 
center  of  mass,  because  the  time-derivative  of  equation  (6)  takes 
the  form 

mf  =  2  /mi'dm  =  mv.  (19) 

Secondly  the  conditions  justify  for  the  next  time-derivative, 

m^  =  2  SJfdm,  (20) 

showing  that  the  center  of  mass  has  the  mass-average  of  accelera- 
tions. Hence  a  particle  having  the  total  mass  (m)  of  the  system 
and  retaining  always  its  position  (f)  at  the  center  of  mass  would 
show  at  every  epoch  the  total  momentum  (Q) ;  and  its  accelera- 
tion would  determine  the  value  (R)  of  the  total  force  in  equation 
(16)  through  the  product  (mv). 

But  if  the  first  terms  in  the  second  members  of  equations  (10) 
and  (12)  and  the  derivatives  of  those  terms  with  regard  to  time 


38  Fundamental  Equations  of  Dynamics 

be  now  considered,  with  the  new  meaning  for  (v),  it  is  seen  that 
the  specified  particle  at  the  center  of  mass  duly  represents  all 
the  dynamical  quantities  for  the  system,  except  those  parts  which 
depend  upon  departures  (r')  from  the  mean  vector  (f)  and  upon 
departures  (u)  from  the  mean  velocity  (v).  Hence  such  an 
artificial  or  fictitious  translation  with  the  center  of  mass  runs  like  a 
plain  thread  through  all  the  equations  for  the  actual  system,  and 
reproduces  accurately  their  six  dynamical  quantities  when  we 
simply  superpose  upon  it  the  additional  kinetic  energy,  moment 
of  momentum,  power  and  force-moment  whose  source  is  in 
the  deviations  from  mean  values.  It  is  a  self-evident  corollary 
that  in  a  real  or  pure  translation  the  particle  at  the  center  of  mass 
represents  the  system  without  corrections,  since  the  local  accelera- 
tions must  be  of  common  value  while  translation  continues,  in 
order  that  simultaneous  velocities  may  remain  equal.  This 
keeps  each  velocity  (u)  permanently  at  zero. 

32.  It  will  be  instructive  to  enforce  without  delay  the  difi"er- 
ences  from  parallelism  with  the  preceding  details  that  appear  at 
several  points,  in  the  simplest  combinations  where  it  becomes 
natural  to  regard  the  total  mass  as  variable  with  time.  Let  us 
then  take  up  for  consideration  a  body  in  translation,  or  equiva- 
lently  a  representative  particle,  denoting  by  (m)  and  (v)  the 
instantaneous  values  of  its  mass  and  velocity.  For  the  momen- 
tum and  the  kinetic  energy  at  the  epoch  we  still  have 

Q  =  mv;        E  =  ^(mvv).  (21) 

If  we  stand  by  the  agreement  that  (Q)  shall  be  force  and  embody 
it  in  the  time-derivative  of  the  first  equation,  we  shall  write 

Q  =  R  =  mt+-^yv.  (22) 

When  mass  is  constant,  resultant  force  and  resultant  acceleration 
have  the  same  direction,  as  we  can  read  in  equation  (16).     But  in 


The  Fundamental  Equations  3^9 

striking  contrast  with  that  consequence,  equation  (22)  shows 
that  its  (R)  does  not  in  general  coincide  with  either  velocity  or 
acceleration. 

Proceeding  next  to  examine  the  power,  and  continuing  to 
specify  it  as  the  derivative  of  (E)  we  find 


dE      _       , ,  ,  .,       1  dm , 

-^  =  P  =  f  (mv-v  +  mv- v)  +  2  "dt"  ^^'"^'^ 


1  dm 


=  mt.v+2-^(v-v).      (23) 

Comparison  with  equation  (22)  brings  out  the  relation 

dm 
dt-^-^ 

dE      1  dm 
=  dt+2-dF^'^-      (24) 


/  dm    \ 

R  •  V  =  I  mv  +  -rr  v  I  •  v  =  m^  •  v  + 


And  once  more  a  variation  from  the  previous  model  is  impressed 
upon  us;  the  power  (P)  is  thrown  out  of  equivalence  with  the 
activity  or  working-rate  of  the  force  (R),  thus  realizing  the 
suggested  contingency  of  section  30.  The  time-integral  of  the 
last  equation  assumes  the  form 


r(R-v)dt  =  mil  +  h  r(dmv.v), 
«/ti  »^t. 


(25) 


and  expresses  on  its  face  the  conclusion  that  the  total  work  of 
the  force  (R)  for  the  interval  is  not  accumulated  in  the  change 
shown  by  the  kinetic  energy.  What  the  form  and  the  fate  are 
of  the  energy  summed  in  the  last  integral  remains  as  a  physical 
question  for  further  study;  it  may,  for  instance,  cease  to  be 
available,  or  it  may  be  stored  reversibly  ready  to  appear  again 
by  transformation. 

If  instead  of  dealing  with  the  resultant  (R)  we  proceed  by  the' 
standard  resolution  into  tangential  and  normal  parts,  these  are 
4 


40  Fundamental  Equations  of  Dynamics 

_  dm  ^  ,     ^ 

R(t)  =  "d^v  +  mt(t);         R(„)  =  mt(n);  (26) 

and  if  we  should  maintain  that  measure  of  force  which  is  ex- 
pressible as  the  product  of  mass  and  its  acceleration,  the  inferences 
from  the  above  equations  would  lead  through  the  quotients  of 
force  by  its  acceleration  to  different  estimates  of  the  mass  in- 
volved.    From  the  first  equation  we  obtain  as  a  ratio  of  tensors 

R(t)      dm  .  .  dv 

-;r^  =  -^v  +  m,         smce        V(t)  =  y^ ;  (27) 

and  from  the  second  equation 

f^  =  m.  (28) 

V(n) 

33.  The  last  value  agrees  with  our  initial  supposition,  and  is 
to  that  extent  the  true  mass ;  and  the  value  given  by  the  first  quo- 
tient in  equation  (27)  has  been  distinguished  as  effective  mass 
since  the  motion  of  a  submerged  body  through  a  liquid  suggested 
the  term.  We  are  aware  how  that  idea  has  been  borrowed  and 
systematized  in  connection  with  the  dynamics  of  electrons;  and 
it  is,  therefore,  of  interest  to  verify  that  the  difference  between 
longitudinal  mass  and  transverse  mass  originally  introduced  there, 
though  now  perhaps  in  course  of  abandonment,  is  quantitatively 
identifiable  with  the  term  (vdm/dv)  according  to  the  assumed 
relations  for  electrons  of  dependence  of  mass  upon  speed. 

The  effect  when  we  are  conscious  of  the  whole  situation  must 
be  to  make  evident  how  much  turns  upon  attributing  the  entire 
force  {R)  to  the  mass  (m),  because  a  force  diminished  by  the 
amount  of  the  last  term  in  equation  (22)  would  reestablish  con- 
formity with  the  type  of  equation  (16)  as 


dm 
d^ 

And  this  is  not  mere  mathematical  ingenuity,  for  in  the  hydro 


(R  -  AR)  =  mv;         AR  =  "^v.  (29) 


The  Fundamental  Equations  41 

dynamical  conditions  at  least  we  know  that  the  excess  of  effective 
mass  over  the  weighed  mass  is  only  a  disguised  neglect  of  back- 
ward force  upon  the  advancing  body  due  to  displacement  of  the 
liquid.  So  that  while  groping  among  phenomena  that  are  less 
understood,  our  attention  should  keep  equal  hold  upon  both 
alternatives  of  statement  until  experimental  analysis  decides 
finally  between  them.  It  is  in  some  degree  a  question  of  words 
whether  all  of  the  force  (R)  falls  within  a  specified  boundary. 

34.  The  formal  changes  that  have  been  pointed  out,  and  their 
possible  reconcilement  with  a  larger  group  of  facts  through  a 
second  phj'sical  view,  are  important  enough  to  justify  this 
immediate  effort  to  fix  attention  upon  them.  The  path  is  beset 
with  similar  ambiguities  whenever  the  details  attendant  upon 
transformations  of  the  subtler  forms  of  energy  are  sought. 
Therefore  it  is  vital  to  pursue  the  thought  of  the  section  referred 
to,  and  to  perceive  with  conviction  even  in  this  simplest  example 
offered,  how  the  bare  assertion  that  a  time  rate  for  mass  will  be 
introduced  for  better  embodiment  of  the  data  leaves  the  dy- 
namics still  impracticably  vague  for  decision.  We  could  not 
pass  upon  the  physical  validity  and  sufficiency  of  the  force  (R) 
assigned  by  equation  (22)  without  fuller  insight  into  suppositions. 
The  instinctive  control  of  the  mathematics  by  repeated  references 
to  the  physics  is  so  well  worth  strengthening  that  we  shall  dwell 
upon  one  other  side  of  the  instance  before  us,  though  for  sug- 
gestion only  and  not  with  any  elaborate  intention  of  exhausting 
it. 

35.  If  a  stream  of  water  flows  steadily  in  straight  stream  fines 
and  with  equal  velocity  everywhere,  there  is  no  loophole  for 
acceleration,  neither  of  an  individual  particle  nor  in  passage 
sj'stematically  from  one  to  another.  Yet  under  an  arbitrary 
agreement  to  include  more  and  more  water  in  the  stipulated 
boundary  the  total  momentum  would  gain  an  assigned  time  rate 
and  the  (R)  of  equation  (22)  a  value 


42  Fundamental  Equations  of  Dynamics 

^       dm  ,     , 

R  =  ^v.  (30) 

This  is  plainly  illusory  and  void  of  dynamical  meaning.  We 
must  cut  off  change  of  mass  by  mere  lapse  of  time;  this  is  one 
wording  of  the  conclusion.  But  on  the  other  hand  conceive  the 
mass  (m)  to  grow  continuously  by  picking  up  from  rest  differ- 
ential accretions,  somewhat  as  a  raindrop  may  increase  by 
condensation  upon  its  surface,  and  equation  (30)  traces  a  phys- 
ical process. 

Investigation  of  this  as  a  physical  action  confirms  equation  (30) 
quantitatively  for  a  proper  surface  distribution  of  the  elementary 
impacts,  as  force  called  for  if  the  slowing  of  speed  is  to  be  com- 
pensated that  would  be  consequent  upon  redistribution  of  the 
same  total  momentum  through  a  continuously  increasing  mass. 
Thus  much  of  force  being  allotted  to  keeping  the  velocity  of  the 
growing  system  constant,  only  the  margin  above  this  part  would 
be  registered  in  the  acceleration.  Moreover  the  way  is  then 
opened  to  interpret  the  last  term  in  equation  (25)  by  adapting 
specially  the  usual  expression  for  kinetic  energy  converted  at 
impact  into  other  forms.  Quoting,  in  a  notation  that  will  be 
understood  at  sight,  we  write  that  loss  in  the  form 

L  =  (l-e^)2-(i;5^j(v.-v.)'.  (31) 

Applying  this  to  the  conditions  of  inelastic  central  impact 
(e  =  0);  with  the  ratio  (mi/m2)  negligible,  as  (dm/m)  is;  and 
when  the  relative  speed  (vi  —  V2)  is  (v) ;  we  find 

L  =  idmv2.  (32) 

And  this  wastage  of  kinetic  energy  finds  due  representation 
through  the  integral  in  question. 

The  essential  condition,  however,  about  (L)  is  a  conversion  of 
kinetic  energy;   and  as  remarked  already  that  conversion  might 


The  Fundamental  Equations  43 

just  as  well  be  reversible.  It  is,  therefore,  again  suggestive  and 
perhaps  even  significant,  that  the  sharing  of  energy  between  two 
forms  indicated  in  the  second  member  of  equation  (25)  can  be 
seen  to  correspond  quantitatively  with  the  partition  of  energy 
between  the  electric  and  the  magnetic  field  of  an  electron  as 
authoritatively  calculated  according  to  the  assumed  rate  of 
change  in  its  mass  with  speed.  Of  course  this  verifies  or  proves 
nothing,  except  the  possibility  in  this  direction  as  in  others  of 
constructing  a  mechanical  process  that  is  quantitatively  adjusted 
to  other  and  different  processes  where  energy  is  converted.^ 

36.  The  six  chosen  quantities  have  been  made  definite  by 
means  of  defining  equations,  which  are  truly  designated  as  funda- 
mental to  the  degree  that  the  quantities  involved  possess  that 
quality.  With  these  identities  we  have  been  content  to  occupy 
ourselves  mainly  thus  far,  and  confine  discussion  to  phenomena 
observed  or  observable  in  a  system  of  bodies,  and  to  be  described 
in  terms  of  the  masses,  their  radius-vectors,  and  two  derivatives 
of  the  latter.  With  data  of  this  type  a  range  of  inferences  can 
be  drawn,  quantitatively  determinate,  too,  up  to  a  certain  point, 
regarding  the  physical  influences  under  which  the  system  will  fur- 
nish those  data.  Any  assumed  local  distribution  of  mass,  velocity 
and  acceleration  demands  calculable  aggregates  of  force,  momen- 
tum and  the  rest,  which  the  equations  can  be  taken  to  specify. 
But  nowhere  along  this  line  of  thought  is  the  further  question 
mentioned,  about  how  the  influences  shall  be  provided  and 
brought  to  bear  in  producing  what  we  see  and  measure,  or  what 
is  visible  and  measurable  in  the  system  that  is  under  observation. 
Not  that  the  relations  prove  finally  to  be  so  one-sided  as  the 
sequence  of  our  mathematics  would  suggest,  according  to  which 
it  happens  that  first  mention  is  given  to  (Q,  E,  H),  and  they  are 
made  primary  in  the  sense  that  the  group  (R,  P,  M)  then  follow 
by  differentiation. 

1  See  Note  11. 


44  Fundamental  Equations  of  Dynamics 

Yet  the  latter  group  would  precede  more  naturally  if  the 
object  were  to  reach  the  first  group  by  integration;  and  this 
inverse  order  is  revealed  to  be  also  a  normal  alternative.  That 
procedure  erects  into  data  the  physical  influences  like  Force, 
Power  and  Force-moment  to  which  the  system  is  externally  or 
internally  subjected,  and  makes  attack  in  the  direction  of  pre- 
dicting the  response  of  the  system  in  detail.  The  unconstrained 
tendency  of  this  line  of  approach  is  then  to  set  forth  the  supple- 
mentary idea  that  the  accumulations  of  Momentum,  Kinetic 
Energy  and  Moment  of  Momentum  in  the  system  of  bodies  are 
to  be  read  as  integrated  consequences  of  the  influences  first 
specified.^ 

The  formal  change  is  inconsiderable,  though  the  spirit  of  it 
guides  three  of  our  announced  identities  into  full-fledged  equa- 
tions either  of  whose  members  is  calculable  in  terms  of  the  other. 
By  usual  title,  these  are  the  Equations  of  Motion,  Work  and  Im- 
pulse that  are  an  important  part  of  dynamical  equipment  and 
rthat  will  next  engage  our  attention.  Since  deciphering  and  list- 
ing the  operative  physical  conditions  comes  now  to-  the  front,  the 
weighing  of  arguments  converges  upon  making  the  list  of  forces 
that  is  sought  exhaustive,  and  upon  weeding  out  illusory  items 
from  it.  It  must  be  apparent  how  that  search  and  critical 
revision  are  bound  up  with  inquiries  Uke  the  suggestions  of  the 
previous  section. 

37.  Dynamical  analysis  of  results  in  its  field  has  everywhere 
made  tenable  and  corroborated  the  thesis  that  momentum  and 
kinetic  energy  are  traceable  as  fluxes.  This  is  understood  to 
imply  that  each  local  increase  of  those  quantities  will  be  found 
balanced  against  some  other  local  decrease,  either  manifest  in 
the  quantity  as  such,  or  finally  detectable  under  certain  disguises 
of  transformation.  In  application  to  a  system  of  bodies,  this 
means  identifying  a  process  of  exchange  dependent  upon  what  is 

1  See  Note  12. 


The  Fundamental  Equations  45 

in  some  sort  external  to  it,  and  sometimes  located  to  occur  over 
the  whole  boundary  or  over  limited  areas  of  it,  or  sometimes 
recognized  to  permeate  the  whole  volume  or  Umited  regions  of  it. 
Under  the  conditions  that  go  with  change  in  total  mass  by  the 
passage  of  material  through  the  stipulated  boundary,  the  mass 
thus  gained  or  lost  may  just  carry  its  momentum  and  kinetic 
energy  out  or  in,  without  any  complicating  interactions. 

If,  however,  we  exclude  and  put  aside  sach  processes  of  pure 
convection  by  confining  ourselves  to  complete  mass-constancy, 
there  is  evidence  that  changes  in  the  total  kinetic  energy  and  the 
total  momentum  of  a  system  of  bodies  are  accompanied  uni- 
versally by  exhibitions  of  force  at  the  seat  of  the  transfer.  And 
this  remains  equally  true  whether  a  transformation  between 
other  recognizable  forms  and  the  mechanical  quantities  denoted 
by  (Q)  and  (E)  is  taking  place  there  or  not.  The  possible  ex- 
changes between  kinetic  energy  and  other  types,  and  the  change- 
ratios  corresponding  to  them  are  a  commonplace  of  modern 
physics;  as  also  we  know  how  refined  measurement  has  attested 
the  forces  upon  bodies  at  transformations  hke  that  into  light- 
energy.  The  settled  anticipations  in  those  respects  have  become 
even  strong  enough  to  look  confidently  upon  occasional  failure 
as  only  postponed  success.  The  more  recent  proposal  is  to  in- 
clude momentum  as  well  as  kinetic  energy  within  the  scope  of 
these  ideas  and  concede  for  both  alike  a  conversion  into  less 
directly  sensible  modifications,  with  force  exerted  upon  bodies 
of  the  system  or  by  them  as  a  symptom  of  the  transformation. 
And  there  seems  to  be  no  cogent  reason  why  this  should  not  hold 
its  ground. 

38.  The  quantitative  formulation  of  these  two  transfers  by 
flux  in  relation  to  what  we  shall  call  the  transfer-forces  tem- 
porarilj'-  and  for  the  purpose  of  present  emphasis  because  they 
are  symptomatic  of  sach  action,  presents  to  us  the  familiar 
equations  of  impulse  and  work  which  shall  be  first  written,  with 
the  usual  mass-constancy  supposed,  in  the  forms 


46  Fundamental  Equations  of  Dynamics 

Q  -  Qo  =  2  j    dR'dt  (The  Equation  of  Impulse);     (33) 
Jo 

E  -  Eo  =  2  f    dR'-ds'  (The  Equation  of  Work).      (34) 

Jo 

They  are  intended  to  express  total  change  from  (Qo)  to  (Q) 
during  any  time-interval  (0,  t),  and  total  change  from  (Eo)  to 
(E)  during  any  simultaneous  displacements  (0,  s')  at  the  points 
of  apphcation  of  the  transfer-forces  (dR').  The  integrations 
then  cover  the  summation  of  effect  over  time  or  distance  for  each 
differential  force  (dR');  and  the  symbol  (S),  though  open  to 
mathematical  criticism  as  a  crude  notation,  is  doubtless  suf- 
ficiently indicative  of  a  purpose  to  include  the  aggregate  of  all 
such  forces  at  every  area  and  volume  where  the  transfers  may 
be  proceeding.  We  must  make  also  the  necessary  discrimination 
between  the  forces  denoted  by  (dR')  and  those  symbolized  by 
(dR)  in  equation  (16),  that  are  localized  by  association  with 
elements  of  mass  and  not  by  participation  in  some  transfer 
process,  and  that  express  themselves  through  the  local  accelera- 
tions manifested  within  the  material  of  the  system  of  bodies, 
while  the  forces  (dR')  can  be  determined  wholly  or  to  an  im- 
portant degree  by  data  extraneous  to  the  system. 

It  should  be  remarked  next  how  one  summation  prescribed 
by  the  second  member  of  equation  (33)  can  be  executed  without 
further  knowledge  or  specification,  since  the  one  time-interval 
applies  in  common  to  all  force  elements  (dR')  that  are  making- 
simultaneous  contributions  toward  the  total  change  (Q  —  Qo). 
Hence  if  the  vector  sum  of  these  forces  in  whatever  distribution 
they  occur  be  written  (R'),  the  explained  sense  of  this  addition 
standing  entirely  in  parallel  with  the  comment  attached  to  (R) 
in  equation  (16),  we  see  that 


Q  -  Qo  =  Tr' 

t/O 


dt.  (35) 


The  Fundamental  Equations  47 

A  corresponding  general  reduction  of  equation  (34)  would  first 
require  equal  vector  displacements  (ds)  at  all  points  of  appli- 
cation throughout  the  group  of  (dR'),  a  condition  that  need  not 
be  satisfied. 

A  second  essential  difference  between  the  equations  of  impulse 
and  work  is  that  the  former  includes  indifferently  every  force 
(dRO,  in  that  some  duration  of  its  action  is  a  universal  charac- 
teristic. But  in  order  that  a  force  (dR')  may  be  effective  in 
work,  not  only  must  there  be  displacement  at  the  point  where 
it  acts  upon  the  system,  but  that  displacement  must  not  be 
perpendicular  to  the  line  of  the  force.  Either  of  these  conditions 
may  be  at  variance  with  the  facts.  It  is  a  convenient  usage  to 
distinguish  transfer-forces  as  constraints  when  they  do  no  work; 
which  signifies  also  when  their  work  is  negligible,  of  course. 

39.  Both  (R')  and  (R)  are  vector  sums  and  have  been  exposed 
in  their  formation  similarly  to  cancellation,  but  there  is  no  pre- 
supposed relation  of  correspondence  in  detail  between  the  two 
groups  that  would  coordinate  the  occurrence  and  the  extent  of 
such  spontaneous  or  automatic  disappearances  from  the  two 
final  totals.  If  however  we  begin  by  confining  comparison  to 
those  totals  as  such,  that  is  yielded  through  the  correlation  of 
two  statements  which  are  now  before  us.  Form  the  time-deriva- 
tive of  equation  (35),  replacing  (Q)  by  its  defined  general  equiva- 
lent from  equation  (I)  and  repeating  its  conditional  derivative 
from  equation  (16).     The  consequence  to  be  read  is 

d  r 

Q  =  R  =—  j    RMt  =  R';  (36) 

and  the  relation  between  the  extreme  members  of  the  equahty 
is  contingent  only  upon  the  vahdity  of  equation  (35).  This 
would  carry  the  equahty  unconditioned  otherwise  of  (R')  and 
(R)  if  (Q  =  R)  can  be  introduced  as  a  defining  general  equation. 
It  gives  latitude  enough  for  the  present  line  of  thought  to  accept 
(R)  as  first  quoted. 


48  Fundamental  Equations  of  Dynamics 

On  its  surface  the  last  equation  offers  the  meaning  that  the 
forces  appUed  to  the  system  under  the  rubric  (R')  are  competent 
to  furnish  exactly  the  total  of  force  exhibited  through  the  con- 
stituents of  (R).  And  the  same  leading  idea  dictates  the  other 
verbal  formula:  The  forces  (dR)  are  an  emergence  of  the  group 
(dR')  after  a  transmission  and  a  local  redistribution.  But  neither 
reading  is  a  truism,  as  the  world  has  realized  since  d'Alembert 
first  made  the  truth  evident;  for  equation  (36)  does  no  more 
than  convey  one  fruitful  aspect  of  d'Alembert's  principle  which 
declares  equality  for  the  impressed  forces  (R')  and  the  effective 
forces  (R),  which  names  sanctioned  by  general  usage  we  shall 
now  adopt,  and  standardize  the  relation  as  the  equation  of  motion 
under  the  form 

2(dR0  =  2  /nivdm.     [The  Equation  of  Motion.]        (37) 

In  the  first  member  the  sign  (S)  recurs  to  the  intention  explained 
for  equations  (33,  34);  and  the  particular  basis  of  the  second 
member  has  been  made  part  of  the  record. 

It  is  already  clear  that  we  have  now  come  to  deal  with  an 
equation  by  whose  aid  can  be  calculated  either  what  total  of 
impressed  force  is  adequate  to  produce  designated  accelerations 
in  given  masses  or  what  distributions  of  accelerations  through- 
out a  mass  are  compatible  with  a  known  group  of  impressed 
forces  as  their  consequence.  But  the  predicated  equality  is 
restricted  to  the  totals  and  contains  that  element  of  indeter- 
minateness  which  affects  every  resultant,  in  so  far  as  it  is  an 
unchanging  representative  of  many  interchangeable  sets  of  com- 
ponents. And  in  any  properly  guarded  terms  that  are  equiva- 
lent to  the  statement  made  above,  the  acknowledged  deduction 
from  the  equality  is  in  its  chief  aspect  a  conclusion  about  the 
acceleration  at  the  center  of  mass  of  the  system  when  (R')  is 
known,  or  a  foreknowledge  of  what  (R')  must  be  somehow  built 
up  if  that  center  of  mass  is  to  be  accelerated  according  to  a 
known  rule. 


The  Fundamental  Equations  49 

40.  If  there  were  complete  physical  independence  among  the 
masses  of  a  system,  or.  in  the  current  phrase,  if  there  were  no 
connections  and  constraints  active  between  them  to  hamper 
mutually  the  freedom  of  their  individual  motions,  impressed 
forces  would  make  their  effects  felt  only  locally  where  they  were 
brought  to  bear.  And  then  for  each  such  subdivision  of  the 
total  mass  as  was  thus  affected  equation  (37)  would  apply,  and 
an  impulse  equation  would  follow.  Observe  however  that  the 
question  of  minuteness  in  the  subdivision  enters,  and  that 
practically  halt  will  be  made  with  some  undivided  unit,  assigning 
to  it  a  common  value  of  acceleration;  so  that  the  center  of  mass 
idea  reappears  in  this  shape  ultimately,  and  duly  proportioned 
to  the  scale  of  force-distribution  symbolized  by  (dR'). 

In  actual  fact  there  are  found  to  be  connections  among  the 
parts  of  a  system  of  bodies,  whose  local  influence  deflects  the 
acceleration  from  being  purely  the  response  to  the  local  quota 
of  (dR').  In  other  words,  the  masses  of  the  system  can  exercise 
upon  each  other  a  group  of  forces  internally,  which  must  be  re- 
garded as  superposed  upon  the  impressed  forces  before  the  account 
of  locally  active  force  is  to  be  held  complete.  To  be  sure  this 
reduces  to  the  now  almost  instinctive  perception  that  external 
and  internal  are  relative  in  use,  and  that  an  action  may  be 
impressed  from  outside  upon  a  part  which  is  exercised  internally 
in  respect  to  a  larger  whole.  But  like  many  other  simple  thoughts 
it  was  once  announced  for  the  first  time. 

Now  certain  forces  being  impressed,  and  with  whatever 
internal  connections  interposed  that  the  system  is  capable  of 
exercising,  the  net  outcome  is  an  observable  group  of  effective 
forces.  It  is  therefore  common  sense  to  conclude  that  this 
net  effect  could  be  entirely  nullified,  in  respect  to  the  accelera- 
tions produced  locally,  by  a  second  group  of  impressed  forces 
applied  also  locally,  and  everywhere  equal  and  opposite  to  the 
local  value  given  by  (dR).     In  virtue  of  equation  (37)  moreover 


50  Fundamental  Equations  of  Dynamics 

it  becomes  apparent  that  the  supposititious  second  group  of  im- 
pressed forces  would  always  amount  in  their  aggregate  to  (—  R'). 
Hence  two  auxiliary  conclusions  can  be  stated:  First  and  nega- 
tively, that  the  superposed  internal  connections  do  not  on  the 
whole  modify  the  original  net  sum  (R'),"  and  the  second  is 
positive,  to  the  effect  that  the  office  of  internal  connections  in 
these  relations  is  to  transmit  and  make  effective  where  they 
would  otherwise  not  be  felt  in  the  system,  the  distribution  of 
impressed  forces  (dR'). 

The  internal  connections  can  be  described  legitimately  as  them- 
selves in  equilibrium;  they  are  the  lost  forces  of  d'Alembert. 
And  the  really  applied  group  (dR')  would  be  in  equilibrium  also 
with  our  second  group  of  locally  impressed  forces.  But  this 
compensation  is  a  supposition  contrary  to  fact;  the  resultant 
(R')  is  unbalanced  force  to  use  the  ordinary  phrase.  These 
details  of  interpretation  are  requisite  exposition  of  the  formally 
insignificant  change  that  writes  instead  of  equation  (37) 

S(dR')  -  2  /mtdm  =  0; 

(38) 
S(dR'  •  5s')  -  S  frai^dm)  •  5s  =  0; 

as  a  formulation  of  d'Alembert's  principle.  The  second  form  in- 
volves the  so-called  virtual  velocities  (5s',  5s),  which  term  is  fairly 
misleading;  for  these  symbols  designate  any  displacements  con- 
sistent with  preserving  the  internal  connections  intact,  and  capa- 
ble of  occurring  simultaneously;  one  group  at  the  driving  points 
of  (dR')  and  the  other  locally  at  each  (dm).  Obviously  either 
form  aims  to  express  that  fictitious  equilibrium  which  is  derivable 
from  the  real  conditions.  Because  the  second  form  is  cast  into 
terms  of  work,  it  seems  to  call  for  the  remark  that  the  founda- 
tion upon  which  all  of  this  is  reared  lies  nevertheless  in  the  im- 
pulse equation,  and  the  development  might  be  called  an  expan- 
sion of  consequences  under  Newton's  third  law;  there  is  no  vital 
bearing  upon  the  actual  energy  relations  definitively  established 


The  Fundamental  Equations  51 

by  it.  What  remains  to  be  said  in  the  latter  respect  we  shall 
next  consider. 

41.  The  first  and  familiar  fact  is  that  the  kinetic  energy  of 
a  system  of  bodies  can  be  affected  by  interactions  that  are  usually 
styled  internal :  quotable  instances  being  gravitational  attraction 
between  sun  and  earth,  and  the  effects  of  resilience  upon  distorted 
elastic  bodies.  Therefore  some  deliberate  caution  must  be 
observed  in  delimiting  the  terms  external  and  internal  in  rela- 
tion to  impressed  forces,  if  equation  (34)  is  to  cover  the  total 
change  in  kinetic  energy  and  yet  make  no  dislocation  from  the 
impulse  equation.  It  will  be  noticed  that  the  critical  instances 
are  connected  with  transformations  of  energy;  and  of  energy 
that  one  mode  of  speech  would  describe  as  internal  to  the  system.^ 
We  can  put  force  exercised  upon  a  body  by  action  of  the  ether- 
medium  into  the  other  category,  since  that  medium  is  by  explicit 
supposition  external  to  our  conception  of  body. 

The  case  of  gravitation  is  resolved  by  the  consideration  that 
the  conversion  of  its  potential  energy  into  the  kinetic  form  is 
attended  with  exercise  of  equal  and  opposite  forces  upon  two 
bodies,  according  to  inference  from  observation.  If  both  bodies 
are  included  in  the  system,  these  forces  cancel  each  other  and  do 
not  disturb  previous  conclusions;  and  if  one  body  is  outside  the 
system's  boundary,  its  action  appears  among  the  (dR').  A 
parallel  statement  can  be  drawn  up  for  elastic  deformations; 
but  there  is  a  remnant  of  combinations  that  are  more  obscure, 
like  the  transformations  of  molecular  and  atomic  energies  that 
can  also  affect  kinetic  energy,  and  that  are  by  common  usage 
attributed  to  the  system  as  an  internal  endowment.  Our 
ignorance  of  their  more  intimate  nature  however  does  not  seem  a 
barrier;  we  can  still  look  upon  every  change  in  a  system's  kinetic 
energy  as  accompanied  by  impressed  forces  (dR'),  whether  these 
are  exerted  in  self-compensated  pairs  and  removed  thus  from 

1  See  Note  13. 


52  Fundamental  Equations  of  Dynamics 

influence  upon  the  impulse  equation,  or  whether  there  are  un- 
balanced elements  that  affect  the  total  momentum  in  addition 
to  changing  the  kinetic  energy.  To  this  extent  all  impressed 
forces  can  be  called  external,  though  there  may  be  hesitation 
about  classing  as  external  or  internal  the  particular  type  of 
energy  that  is  under  transformation  to  or  from  the  kinetic  form. 
The  corollary  may  be  added,  that  so  long  as  equal  and  opposite 
elements  of  force  are  also  colinear,  their  moments  for  any  origin 
are  self-cancelling;  otherwise  they  constitute  couples. 

With  the  attempt  to  formulate  correct  equations  of  motion, 
the  difiiculties  of  physical  dynamics  may  be  said  to  begin,  when 
it  is  required  to  make  the  list  of  impressed  forces  what  we  have 
spoken  of  as  exhaustive  and  freed  from  illusions.  Outside  the 
range  of  rather  direct  perceptions,  we  grapple  with  uncertainties 
under  conditions  of  imperfect  knowledge — with  hypothetical 
forces,  intangible  energies,  figurative  masses.  Dynamics  that 
was  ready  to  renounce  criticism  of  provisional  equations  of 
motion  would  be  over-sanguine.  Conversions  of  energy  into  the 
one  distinctively  mechanical  form  that  we  call  kinetic  are  perhaps 
closest  to  direct  inquiry  into  attendant  circumstances;  and 
though  it  would  be  overcautious  to  construct  on  that  base  only, 
it  seems  probable  that  dissecting  there  first  is  the  clew  to  larger 
success,  and  that  equations  (33,  34,  36)  are  landmarks  on  that 
road. 

In  practice,  the  bare  statement  of  d'Alembert's  principle  as 
given  by  any  one  of  the  three  forms  indicated  is  supplemented 
with  some  record  of  the  particular  connections  that  overcomes 
the  difiiculty  of  specifying  every  individual  local  acceleration, 
and  reduces  the  number  of  indispensable  data  within  manageable 
limits.  The  forces  of  the  connections  are  thus  described  in- 
directly through  the  geometrical  equations  of  condition;  and  this 
method  is  more  effective  than  the  more  direct  one,  because  the 
magnitude  of  the  constraining  forces  will  in  general  depend  upon 


The  Fundamental  Equations  53 

the  speeds,  though  the  kinematical  analysis  of  the  hnkages 
remains  unaltered.  It  is  this  thought  that  introduced  Lagrange's 
use  of  indeterminate  multipliers} 

None  of  these  devices  though  qualifies  the  character  of  d'Alem- 
bert's  equality  in  asserting  a  quantitative  equivalence  between 
a  net  total  of  external  agency  (impressed  forces)  and  the  response 
to  it  on  the  part  of  a  system  of  bodies,  as  expressed  in  the  states 
of  motion  that  the  effective  forces  summarize.  The  physical 
thought  attaching  to  the  equation  of  motion  will  be  clearer 
when  cause  and  effect  are  kept  apart,  and  will  tend  toward 
obscurity  or  confusion  when  a  shuffling  of  terms  from  one  member 
to  the  other,  as  a  mathematical  device  or  for  other  reasons,  has 
impaired  this  desirable  homogeneousness. 

42.  One  large  section  of  dynamics  is  devoted  to  working  out 
its  principles  in  their  application  to  rigid  solids.  As  these  are 
specified,  they  carry  to  an  extreme  limit  a  scheme  of  inter- 
connections among  their  constituent  parts  that  provides  an 
ideal, of  internal  structure  which  knows  no  rupture  nor  even 
distortion,  but  which  provides  inexorably  all  necessary  con- 
straining connections.  Like  other  such  concepts  its  considera- 
tion yields  results  which  are  not  only  valuable  in  themselves, 
but  which  also  furnish  a  point  of  departure  for  the  introduction 
of  conditions  that  approach  their  standards  closely  enough  to 
be  taken  account  of  by  means  of  small  corrective  terms.  Beside 
repeating  that  frequent  and  useful  relation  of  a  concept  to  actual 
data,  the  study  of  rigid  dynamics  has  some  more  special  reasons 
to  support  it,  of  which  one  is  discoverable  in  the  trend  of  the- 
oretical views  about  the  constitution  of  all  systems  of  bodies. 
The  boldest  analysis  of  molar  and  molecular  and  atomic  units, 
as  a  substratum  for  the  increasing  number  of  energy-forms  that 
we  associate  with  them  and  give  passage  through  them,  has  not 
broken  away  entirely  from  utilizing  rigid  solids  of  smaller  scale 

*  See  Note  14. 


54  Fundamental  Equations  of  Dynamics 

and  their  dynamics.  This  gives  the  prevailing  tone  in  attacking 
the  atomic  nucleus  and  its  atmosphere  of  electrons  even,  with 
only  such  mental  reactions  to  modify  the  trust  in  the  details  of 
the  reasoning  as  have  a  wholesome  influence  to  maintain  the  flexi- 
bihty  that  is  scientific  and  make  our  dynamics  more  nearly 
universal  in  what  it  embraces.^  In  this  sense  the  kinematical 
phase,  through  which  so  many  of  these  matters  evolve,  remains 
uncompleted — or  we  may  dub  it  empirical — until  dynamics  can 
serve  it  with  reasoned  argument. 

In  the  second  place,  however,  any  rule  of  constancy  is  likely 
to  have  an  advantage  of  particular  kind  over  the  multifarious 
rules  of  variation  in  correlation  with  which  it  is  unique.  This 
goes  beyond  the  formal  gain  in  abohshing  some  mathematical 
complications,  though  that,  too,  frees  our  minds  to  entertain  the 
salient  ideas  with  fuller  concentration.  Like  our  previous 
assumption  of  constancy  in  mass,  this  added  supposition  of 
permanent  internal  arrangement  puts  off  particularizing  among 
rules  of  change,  and  enables  us  to  carry  forward  through  instruc- 
tive developments  the  task  of  bringing  some  general  principles 
more  nakedly  to  discussion.  This  grows  cumbrous  or  impossible 
where  conclusions  are  subject  to  many  contingent  decisions. 

43.  It  bears  rather  closely  upon  these  suggestions  that  we  can 
make  one  good  entry  upon  the  particular  inquiries  about  rigid 
solids  by  resuming  and  continuing  the  hne  of  thought  that 
paused  at  equation  (20).  In  that  section  some  glimpses  were 
secured  of  a  superposition  by  means  of  which  a  serviceable  sketch 
can  be  drawn  of  a  dynamical  outline  for  certain  systems  of  bodies. 
Or  otherwise  stated,  the  actual  totals  of  the  important  quantities 
are  grouped  round  the  concept  of  a  representative  particle,  leav- 
ing only  specified  remainders  for  further  consideration.  Let  us 
now  separate  from  such  a  system  one  body  that  we  shall  suppose 
rigid  and  having  continuous  mass-distribution,  and  deduce  for 

1  See  Note  15. 


The  Fundamental  Equations  55 

it,  with  increased  finality  of  detail,  the  special  consequences  that 
seem  valuable  for  our  purpose.  It  is  clear  that  the  center  of 
mass  of  this  body  will  retain  all  the  functions  already  assigned 
to  the  representative  particle,  and  also  that  it  must  now  in 
addition,  because  the  body  is  rigid,  fall  into  an  unchanging 
configuration  that  makes  constant  in  length  all  such  vectors  as 
(r')  of  equation  (12).  And  it  follows  too  from  the  conception  of 
rigidity  that  the  internal  connections  are  excluded  from  net 
effect  upon  the  sequences  of  conversion  that  change  the  body's 
kinetic  energy.  They  are  reduced  in  their  final  influence  to  the 
office  of  transmitting  and  distributing  the  consequences  of  con- 
versions and  constraints  that  have  been  effected  otherwise  than 
by  any  machinery  of  readjustments,  named  or  unnamed,  of  in- 
ternal arrangement.  The  intended  meaning  is  not  essentially 
varied,  though  it  has  been  rendered  less  explicit  perhaps,  when 
it  is  said  that  the  impressed  forces  can  here  only  displace  the  body 
as  a  whole,  or  that  the  internal  connections  can  do  no  work. 

44.  Now  it  is  the  elementary  characteristic  of  translation  that 
it  does  apply  to  the  body  as  a  whole  and  affect  it  uniformly 
throughout  in  all  kinematical  respects.  Our  next  natural  step, 
therefore,  is  to  examine  the  remaining  possibiHty  that  is  con- 
sistent with  the  constant  length  of  every  (r'),  and  that  therefore 
restricts  the  locus  of  each  mass-element  to  some  sphere  that  is 
centered  on  the  center  of  mass.  If  we  accept  for  this  type  of 
motion  as  a  whole  the  term  rotation,  there  still  remain  some 
particulars  to  establish  definitely;  and  of  these  the  first  will  be 
the  general  value  of  the  velocity  denoted  by  (u)  in  equation  (9), 
for  which  one  fitting  name  is  the  local  velocity  relative  to  the  center 
of  mass.  It  is  evidently  identical  with  the  local  velocity  (v)  of 
each  (dm)  if  (v)  is  zero,  or  if  the  center  of  mass  is  the  origin  of 
reference.  With  control  of  the  value  for  (u)  we  can  ultimately 
take  up  the  evaluation  of  the  terms  that  contain  (u)  or  depend 
upon  it,  knowing  in  advance  that  these  can  appear  in  (E,  H, 
P,  M)  but  not  in  (Q,  R). 
5 


56  Fundamental  Equations  of  Dynamics 

45.  In  order  to  approach  the  matter  conveniently  let  (CO 
denote  the  center  of  mass,  and  locate  orthogonal  axes  there  that 
are  lines  of  the  body:  that  is,  they  move  with  the  body  and  retain 
their  positions  in  it.  The  unit-vectors  of  those  axes  shall  be 
(i',  y,  k')  in  the  standard  right-handed  cycle.  Then  using  the 
word  temporarily  in  an  untechnical  sense,  any  rotation  relative 
to  (C)  will  in  general  change  all  the  angles  that  (i',  j',  k')  make 
with  the  reference-axes.  Consider  first  differential  changes  of 
orientation  (o,  g,  5)  matching  the  order  of  the  unit- vectors. 
Then  (a)  as  an  angle-vector  is  normal  to  the  plane  of  the  con- 
secutive positions  of  (i');  similarly  for  (g)  and  (j'),  and  for  (5) 
and  (k').  The  corresponding  linear  displacements  on  unit 
sphere  around  (C)  are  given  as  products  of  perpendicular  factors 

by 

di'  =  a  X  i';        dj'  =  5  X  j';        dk'  =  5  x  k'.         (39) 

The  vector  products  are  not  affected,  and  hence  these  equalities 
are  not  disturbed,  if  we  introduce  three  arbitrary  elements  of 
angular  displacement;  (X')  in  the  line  of  (i')  into  the  first,  (i»') 
in  the  line  of  (j')  into  the  second,  and  (v')  in  the  line  of  (k')  into 
the  third,  writing 

di'  =  (a  +  ^Oxi';        df  =  (5  +  v')xj';  ^    ^ 

(40) 
dk'  =  (5  +  v')  X  k'. 

But  because  the  axis-set  must  remain  orthogonal  in  the  rigid 
body,  the  elements  of  angular  displacement  in  the  line  of  the 
third  axis  must  always  be  equal  for  the  two  other  axes  at  the 
same  stage.  This  renders  possible  the  adjustments  of  particular 
values  that  make  equations  (40)  simultaneous: 

^  =  5(i')  =  S(i');       v  =  a(j')  =  6(j'); 

(41) 

V  =  o(k')  =  5(k'); 

with  the  consequence  that  equations  (40)  are  satisfied  in  the 
forms 


The  Fundamental  Equations  57 

di'  =  dy  X  i';        dj'  =  dr  X  j';        dk'  =  dy  x  k'; 

(42) 

dy  =  ^  +  V  +  V. 

The  occurrence  of  the  vector  (dy)  as  a  common  factor  in  all  three 
equations,  combined  with  its  determination  by  projections  on 
axes  arbitrarily  chosen  and  with  the  fact  that  simultaneous 
linear  displacements  at  points  in  the  same  radius-vector  must 
be  proportional  to  distances  from  (C),  shows  that  at  each  epoch 
and  for  every  (r')  of  constant  length, 

dr'  =  dy  X  r';        f '  =  u  =  &>  x  r';        u  =  y.  (43) 

Here  («)  denotes  the  rotation-vector  for  either  body  or  axis-set, 
of  course,  since  they  are  supposed  to  turn  together.  It  follows 
without  further  question  that  if  a  rigid  sohd  moves  so  that  all 
its  radius-vectors  (r)  measured  from  any  reference-origin  remain 
of  constant  length,  the  simultaneous  velocities  (v)  of  all  mass- 
elements  conform  to  the  relation 

V  =  fa)  X  r.  (44) 

Any  such  motion  as  a  whole  is  described  as  a  pure  rotation  with 
angular  velocity  (&>),  for  which  vector  the  origin  is  conventionally 
the  base-point. 

46.  The  vector  (w)  is  usually  termed  the  angular  velocity  of 
the  body  at  the  epoch,  the  phrase  being  made  reasonable  by  the 
appearance  of  (fa))  as  a  factor  common  to  all  radius-vectors  in 
equations  hke  (43)  or  (44).  But  both  the  procedure  by  which 
this  angular  velocity  was  determined  and  its  appearance  in  a 
vector  product  show  plainly  that  its  resultant  value  is  not 
effective  to  produce  changes  of  direction  in  all  radius-vectors.^ 
This  common  factor  has  been  seen  to  include  three  elements 
that  become  superfluous  each  for  one  axis,  as  not  influencing 

1  See  Note  16. 


58  Fundamental  Equations  of  Dynamics 

angular  displacement  of  it,  nor  the  corresponding  linear  displace- 
ment of  points  in  it.  The  rotation- vector  is  thus  open  to  inter- 
pretation as  a  maximum  value,  useful  in  giving  through  its  pro- 
jection upon  the  normal  to  any  plane  at  its  base-point  the  part 
effective  to  bring  about  a  complete  angular  displacement  oc- 
curring in  that  plane.  If  we  identify  (o>)  with  the  line  of  a  rota- 
tion-axis, permanent  or  instantaneous,  these  explanations  are 
consistent  with  the  elementary  ideas  of  spin  about  the  rotation- 
axis  and  hnear  velocity  given  by  the  product  of  rate  of  spin  and 
distance  from  the  axis. 

47.  The  preceding  identification  of  a  rotation-vector  connects 
its  considerations  with  departures  from  configurations  of  (i'j'k') 
that  are  themselves  subject  to  self-produced  change,  in  so  far 
as  they  move  with  the  body;  and  this  might  conceivably  modify 
the  result.  But  if  that  loop-hole  seems  to  exist  it  is  closed  when 
we  detect  the  same  vector  (dy)  in  direct  terms  of  its  projections 
upon  the  reference-axes  oriented  by  (ijk)  permanently.  And 
it  is,  further,  worth  while  to  do  that,  because  these  projections 
are  uniquely  advantageous  in  preparing  for  algebraic  additions 
to  express  any  resultant  angular  displacement  according  to  the 
relation 

Y  =  fdy  =  i/d7(i)  +  j/d7(j)  +  k/d7(k),  (45) 

the  tensors  that  are  integrated  being  those  of  the  projections  of 
each  (dy)  upon  the  axes  of  (i,  j,  k).  The  confirmation  sought 
depends  upon  satisfying  the  relations, 


d7(i)  =  Xi'-i  +  MJ'-i  +  "k'-i 
d7(i)  =  Xi'-j  +MJ'-J  +  "k'-j 
d7(k)  =  Xi'-k  +  MJ'-k  +  I'k'-k. 


(46) 


Ordinary  routine  verifies  that  equations  (46)  fulfil  identically 
the  necessary  conditions: 


The  Fundamental  Equations  69 

di'  =  dy  X  i'  =  i(i'(k)d7(i)  -  i'(j)d7(k)) 

+  j(i'(i)d7(k)  -  i'(k)d7(i)) 

+  k(i'(nd7(i)  -  i'(i)d7(i)).  r    (47> 

dj'  =  dy  X  j'  =  etc. 

dk'  =  dy  X  k'  =  etc. 

It  is  not  without  interest  to  notice  in  detail  how  algebraic  cancel- 
lations now  preserve  the  obligatory  independence  of  (^)  in  the 
results  for  (di') ;  of  (v)  in  those  for  (dj') ;  and  of  (v)  in  those  for 
(dk').  This  second  development  is  more  circuitous,  because 
the  permanently  orthogonal  condition,  due  to  rigidity,  pertains 
intimately  to  (i',  j',  k'),  the  coincidence  of  results  by  both  attacks 
being  a  special  instance  under  a  general  theorem  that  will  be 
proved  subsequently  (see  section  85).  The  equal  corroboration  of 
equation  (44)  is  a  plain  inference,  and  hence,  wherever  a  rotation- 
vector  covers  the  local  velocities  of  a  rigid  body,  or  the  body  is 
in  pure  rotation  about  a  fixed  point,  the  summed  projections  are 
invariant : 

0)(i)   +  W(j)   +  fa>(k)    =   0>(i')   +  0>(j)'  +  G)(k')    =   w.  (48) 

Substitute  in  equation  (44),  use  the  standard  relation  for  common 
origin, 

r  =  X  +  y  +  z  =  x'  +  y'  +  z',  (49) 

and  omit  products  of  colinear  factors.     This  yields 

V  =  fa>(i)  X  (y  +  z)  +  w(i)  X  (z  +  x)  +  fa>(k)  X  (x  +  y)  ] 

\  (50) 
=  w(i')  X  (y'  +  z')  +  a)(j')  X  (z'  +  x')  +  (O(k')  X  (x'  +  y'),  J 

and  is  the  foundation  for  a  standard  rule:  Linear  velocities  in  a 
rotating  rigid  body  are  given  correctly  by  superposing  those  due 
to  separate  partial*  rotations,  either  about  the  reference-axes  or 
about  the  positions  at  the  epoch  of  any  three  lines  of  the  body 
intersecting  orthogonally  at  the  origin. 


60  Fundamental  Equations  of  Dynamics 

48.  In  the  present  connection  however  we  are  deaUng  with 
a  rotation  relative  to  (C)  as  superposed  upon  the  concept 
of  a  representative  particle  and  supplementing  the  latter,  with  a 
proved  equivalence  of  translation  and  rotation  thus  combined 
in  replacing  the  most  general  group  of  velocities  in  our  rigid 
body.  On  incorporating  these  recent  restatements  into  equa- 
tions (10)  and  (12),  they  take  on  the  more  special  forms  that 
we  can  now  exhibit.  Denote  the  last  terms  in  the  two  equations 
by  (Er)  and  (Hr),  which  we  shall  call  briefly  the  kinetic  energy 
and  the  moment  of  momentum  relative  to  the  center  of  mass. 
Then  for  the  one  body  of  continuous  mass 

Hr  =  /ni(r'  X  udm)  =  /xn(r'  x  (w  x  r')dm) 

=  /„(o(r'.r')  -r'(G>.r'))dm;     (51) 

Er  =  I  /mU-udm  =  I  /m((o  X  r')  •  (ui  x  r')dm 

=  I  /m((corO^  -  (co.rO^)dm  =  K^'Hr);     (52) 

the  final  reduction  of  (Er)  being  readily  verifiable,  when  we 
remember  that  (<o)  is  common  to  all  elements  in  these  mass- 
summations. 

49.  Next  we  continue  into  equations  (17)  and  (18)  the  same 
plan  of  partition  between  representative  particle  and  supple- 
mentary term.  Direct  substitution  there  according  to  the  rela- 
tions previously  used, 

V  =  V  +  u;        r  =  f  +  r';  (53) 

gives 

P  =  ^  =  v-R  +  /„u-dR;  (54) 

M  ^  H  =  (f  X  R)  +  /„(r'  X  dR).    .  (55) 

We  may  remind  ourselves  that  the  first  terms  in  the  final 
members  of  both  these  equations  are  in  harmony  with  the  time- 
derivatives  of  corresponding  terms  in  equations  (10)  and  (12) 


The  Fundamental  Equations  61 

if  we  bear  in  mind  equation  (20) ;  and  they  show  how  the  particle 
can  be  reUed  upon  still  to  present  these  contributions  to  power 
and  to  force-moment  as  based  upon  its  artificial  translation  with 
the  center  of  mass.  Denote  the  additional  power  and  force- 
moment  by  (Pr)  and  (Mr);  then  from  equations  (54,  55), 

Pr  =  /m(o>  X  r')  -dR  =  io-fUr'  x  dR)  =  wMr;         (56) 
M^^Uir'xdR).  (57) 

We  shall  compare  these  statements  with  the  consequences  of 
■equations  (51,  52),  which  give  for  their  derivatives 

^(Er)  =K"-Hk  +  <o.Hr);  (58) 

Hk  =  ^  /m(r'  X  udm)  =  /„,(r'  x  lidm) ;  (59) 

because  (u)  and  (r')  are  identical.  Further,  since  differentiation 
of  equation  (9)  shows 

V  -  -^  -f-  u,  (60) 

a  natural  name  for  the  last  term  is  the  local  acceleration  relative 
to  the  center  of  mass,  which  would  indicate  also  a  local  force- 
element  (lidm)  differing  from  (♦dm)  that  is  (dR)  and  thereby 
breaking  the  equality  of  (H^)  and  (M^).     But  since 

/„.r'dm  =  0,         (/„.r'dm)  x  v  =  /„,(r'  x  vdm)  =  0;     (61) 

and  this  term  can  be  added  without  error  to  equation  (59), 
giving     . 

Hr  =  fm{i'  X  (^  +  ii))dm  =  Mr'  x  dR)  =  Mr.        (62) 

Evidently  the  value  in  equation  (61)  could  reversely  be  sub- 
tracted without  error  from  equation  (57).  The  interchange- 
ableness  of  these  forms  should  not  be  lost  sight  of. 

50.  A  similar  concordance  of  equations  (56,  58),  though  it  is 


62  Fundamental  Equations  of  Dynamics 

not  superficially  evident,  follows  at  once  on  showing  a  right  to 
add  the  third  member  in  the  equality 


(*) 


Mr    =    0)-Hr    =    (0-Hr,  (63) 


whose  first  and  second  members  are  now  known  to  be  equal. 
The  required  proof  is  got  by  differentiating  equation  (51),  where 
we  find 

Hr  =  /„,{6(r'-r')  -  u(a>.r')  -  r'(<bT')}dm,  (64) 

whose  scalar  product  with  (&>)  is,  omitting  everywhere  scalar 
products  of  perpendicular  factors, 

o>-Hr  =  /„,{(G>-u))(r'T')  -  (G)T')(6)T'))}dm 

(65) 
=  /mw-(fa>(r'-r')  —  r'(o>-r'))dm  =  o-Hr. 

The  vector  (<b)  which  is  the  time-derivative  of  the  rotation- 
vector  (w)  is  named  the  vector  of  angular  acceleration.  Of  course 
it  provides  for  both  changes  of  direction  (or  of  axis)  in  the  rota- 
tion, and  for  changes  in  its  magnitude  (or  spin) ;  and  (tb)  must 
be  of  common  application  at  any  epoch  to  all  mass-elements, 
because  that  is  true  for  («). 

51.  With  the  support  of  equations  (51,  52,  56,  58),  we  have 
given  .consideration  to  all  four  quantities  that  need  specifjdng, 
for  the  rotation  that  is  the  remainder  over  and  above  the  fic- 
titiously segregated  translation,  since  the  representative  particle 
as  it  has  been  determined  engages  the  totals  of  force  and  momen- 
tum. And  having  brought  the  discussion  to  this  point,  in  terms 
connected  with  the  effective  forces  whose  resultant  is  (R),  it 
remains  to  make  that  transition  to  impressed  forces  with  equal 
resultant  (R'),  which  we  have  learned  to  associate  with  d'Alem- 
bert's  name.  Under  the  conditions  explained  for  rigid  bodies, 
certain  sources  of  impressed  force  are  not  to  be  permitted,  but 
the  total  work  done  must  appear  in  the  energies  of  translation 
and  rotation.     Let  us  then  next  summarize  how  matters  stand 


The  Fundamental  Equations  63 

with  the  six  dynamical  quantities,  in  the  two  groups  that  we 
have  recognized. 
I.  Translation: 

1.  Force  (R'  =  R)  at  (f). 

2.  Momentum  (Q  =  mv)  at  (f). 

3.  Energy  (Et  =  ^mv^). 

4.  Moment    of    Momentum     (H-r  =  f  x  mv) ;      consistent 

with  (2). 

5.  Power   (P^  =  R'-v  =  (d/dt)(ET));    consistent  with    (1) 

and  (3). 

6.  Force-moment    (M^  =  f  x  R'  =  Hx) ;     consistent    with 

(1)  and  (4). 
II.  Rotation: 

1.  Force  =  0;     consistent    with    couples    expressing    self- 

compensating  elements  in  (R'). 

2.  Momentum  =  0  always;  consistent  with  impulse  of  zero 

force. 

3.  Energy  (Er  =  ^wHr). 

4.  Moment    of    Momentum    (Hj^);     consistent   with   zero 

momentum. 

5.  Power  (P^  =  wMr  =  {d/dt)(EjJ);    consistent  with  (1) 

and  (3). 

6.  Force-moment    (Mr  =  H^);     consistent    with    (1),    (3) 

and  (5). 
52.  The  review  of  these  details  impresses  the  fact  that  the 
above  conventional  separation  accomplishes  complete  inde- 
pendence for  two  such  constituents  of  the  actual  data,  in  the  sense 
that  the  course  of  events  can  be  duly  expressed  for  each  group, 
with  indifference  to  the  presence  or  absence  of  the  other,  by  a  self- 
contained  use  of  the  general  dynamical  scheme.  This  secures 
the  full  simplicity  attendant  on  pure  superposition,  by  shrewdly 
exploiting  center  of  mass  for  its  average  properties,  and  kinetic 
energy  with  moment  of  momentum  for  their  salvage  of  what  the 


64  Fundamental  Equations  of  Dynamics 

mean  values  sacrifice,  utilizing  also  a  form  of  Poinsot's  allowance 
through  a  couple  for  off-center  action  of  a  force.  The  idea  is 
successful,  besides,  in  concentrating  into  the  rotation  elements 
where  the  form  and  the  mass-distribution  of  the  bod}^  complicate 
the  data  with  differences;  and  this  frees  the  translation  for  giving 
expression  to  broad  traits  of  similarity. 

The  rudiments  of  the  steps  now  taken  are  perceivable  in  equa- 
tions (10)  and  (12),  where  it  is  plain  that  an  internal  energy  like 
(Er)  could  belong  to  radial  pulsations  of  mass-elements  about 
(C),  either  alone  or  added  to  spin  as  a  whole;  but  development 
is  checked  until  (u)  is  particularized  in  its  value  and  distribution. 
It  is  plain,  however,  that  adaptation  to  many  combinations  is 
feasible,  whose  general  feature  is  non-appearance  in  translational 
energy  of  full  equivalent  for  the  total  work  done.  Failing 
definite  knowledge  that  forbids,  a  rotation  can  be  devised  as  one 
possible  means  of  absorbing  a  quota  of  kinetic  energy,  and  as 
one  guide  to  conjecture  among  the  facts  of  an  observed  diversion 
of  energy  from  a  translation.  It  is  scarcely  necessary  to  insist 
that  the  equivalence  of  any  such  devices  is  restricted  to  those 
particulars  according  to  which  their  lines  were  laid  down;  the 
particle  plus  a  rotation  is  an  equivalent  for  the  general  motion 
of  a  rigid  body  only  in  the  six  respects  enumerated.^ 

53.  At  equation  (44)  the  idea  was  introduced  that  pure  rota- 
tion of  a  rigid  body  about  a  reference-origin,  instead  of  the  center 
of  mass,  is  describable  in  corresponding  terms  on  substituting 
(r)  for  (r')  and  (v)  for  (u).  The  intrinsic  difference  lies  in  the 
necessity  that  a  reference-origin  is  a  fixed  point,  whereas  the 
possible  velocity  of  the  center  of  mass  runs  like  a  thread  through 
all  our  recent  discussion.  Let  us  realize  that  the  main  results 
now  added  can  be  similarly  extended,  and  put  down  as  applicable 
to  pure  rotation  about  the  reference-origin  these  parallels  specif- 
ically to  equations  (51,  52,  56,  62,  65): 

I  See  Note  17. 


The  Fundamental  Equations  65 


H  =  /^^((o(^•^)  -  r(<o-r))dm; 

P  =  coM; 

H  =  M; 

H  =  coH  =  G)M. 


(66) 

Total  quantities      (67) 

for  pure  rota-      (68) 

tion.  (69) 

(70) 

Since  in  this  ease  supposed,  the  center  of  mass  need  not  coincide 
with  the  origin,  the  alternative  choices  will  be  open  to  treat  the 
body  as  exhibiting  rotation  alone,  or  as  affected  with  translation 
and  with  a  rotation  besides.  But  translation  cannot  bring  in 
change  of  direction  for  Hnes  of  the  body,  hence  both  views  of 
the  rotation  must  agree  in  their  rotation-vectors  permanently. 
And  because  the  center  of  mass  cannot  change  its  position  relative 
to  its  rigid  body,  a  relation  distinctive  of  pure  rotation  must  be 

V  =  Q  X  f.  (71) 

The  comparative  directness  and  convenience  of  the  two  methods 
will  be  decided  according  to  circumstances.  One  method  ex- 
cludes from  (M)  any  forces  really  acting  through  the  origin; 
the  other  can  omit  from  (M^)  any  forces  acting  through  (C)- 

54.  We  proceed  with  the  requisite  analysis  of  rotation,  by 
examining  the  specialized  values  of  local  accelerations  and  some 
consequences  of  them,  conscious  always  in  the  light  of  what  has 
just  been  said,  that  the  conclusions  will  be  available  for  twofold 
use.  One  is  more  important,  doubtless,  because  more  inclusive 
in  application  to  the  most  general  type  of  motion  of  which  a  rigid 
body  is  capable;  but  the  second  has  weight,  too,  in  attacking  the 
conditions  of  pure  rotation  that  are  made  prominent,  for  in- 
stance, in  common  forms  of  the  gyroscope. 

The  local  acceleration  of  a  pure  rotation  given  by  differentiating 
equation  (44)  is 

v  =  ((oxr)  +  (toxv).  .  (72) 

Let  us  make  this  form  our  text  and  starting-point,  remembering 
that  in  the  other  circumstances  it  is  to  be  recast  into 


66  Fundamental  Equations  of  Dynamics 

li  =  (g)  X  r')  +  (o>  X  u),  (73) 

with  continuations  where  (r')  replaces  (r)  everywhere  and  (u) 
replaces  (v),  while  (u)  is  read  the  local  acceleration  of  the  rotation 
and  is  the  excess  of  (t)  over  (v).  The  vector  (w)  gives  the 
velocity  of  the  extremity  of  (w),  of  course;  and  its  base-point 
will  be  taken  conventionally  at  the  origin  with  which  our  idea  of 
rotation  is  associated.  Then  the  process  modifying  (w)  by  (d)) 
is  one  of  continuous  parallelogram  composition  for  intersecting 
vectors,  though  equivalent  indeed  to  addition  in  a  triangle. 

The  vector  sum  in  equation  (72)  deserves  close  attention, 
because  though  the  two  types  of  its  terms  are  on  one  count  an 
incident  of  the  algebra,  it  happens  that  they  conform  remark- 
ably, first  to  the  kinematical  elements,  and  later  to  a  certain 
plane  of  cleavage  in  the  djmamics.  The  form  of  the  second  term 
connects  it  conclusively  with  change  of  direction  only  for  its 
velocity;  and  the  first  term  enters  and  vanishes  with  angular 
acceleration.  If  (w)  retains  direction  (w)  must  be  colinear  with 
it;  and  then  first  inspection  can  identify  the  terms  with  the 
tangential  and  the  normal  acceleration  respectively  of  the  local 
(dm)  in  its  circle  perpendicular  to  (w).  But  the  complete  separa- 
tion of  changes  in  magnitude  and  in  direction  for  (v)  that  then 
exists  should  not  be  assumed  more  generally;  it  is  always  true, 
however,  that  the  first  term  in  the  acceleration  bears  the  same 
relation  to  the  axis  of  angular  acceleration  (w)  that  the  corre- 
sponding velocity  (v)  does  to  the  axis  of  rotation  (w). 

55.  Multiplying  equation  (72)  by  (dm)  yields  the  effective 
force-element,  which,  because  it  is  exhibited  locally,  must  have 
a  moment  to  be  found  by  taking  that  force  in  vector  product 
with  its  (r).  The  total  moment  then  demanded  by  the  localized 
forces  must,  as  we  have  seen,  be  furnished  by  the  impressed 
forces;  and  this  amount  is  expressed  by  the  integral 

M  =  /,„[r  X  ((a>  X  r)  +  (to  X  v))dm].  (74) 


The  Fundamental  Equations  67 

Denote  the  two  main  constituents  of  this  moment  by  (M')  and 
(M");  and  let  us  take  up  the  second  part  for  examination. 
Expand  the  triple  vector  product,  omit  the  scalar  product  of 
perpendicular  factors,  and  finally  write  for  (v)  its  known  value. 
This  shows 

M"  =  —  /mV(G)-r)dm  =  —  /^(o  x  r)(fa)-r)dm.  (75) 

Next  form  for  comparison  the  product 

w  X  H  =  /ni(cd  X  [w(r-r)  —  r(wr)]dm) 

=  - /m(w  xr)(o>-r)dm,      (76) 

and  we  see  that  the  extreme  members  are  identical.  Hence  we 
conclude  that  the  office  of  thus  much  of  the  force-moment  is  to 
produce  a  change  of  direction  in  the  vector  of  total  moment  of 
momentum  so  regulated  that  the  latter  would  move  with  the 
body  or  retain  its  position  in  the  bodj'.  This  is  a  simple  corollary 
of  the  interpretation  of  (g>)  according  to  section  (47).  If  ((,i) 
and  (H)  were  in  every  case  colinear,  their  vector  product  at  the 
value  zero  would  become  formal  and  meaningless.  But  it  appears 
plainly  in  equation  (66),  first  that  (H)  may  be  thrown  out  of 
line  with  («)  by  the  term 

—  /mr(wr)dm, 

which  does  not  in  fact  generally  vanish  nor  become  colinear 
with  (w);  and  secondly,  that  (H)  and  (to)  cannot  become  per- 
pendicular by  compensations  within  the  first  term,  because  every 
product  (r-r)  is  essentially  positive.  That  they  never  are  perpen- 
dicular we  shall  conclude  presently  (see  section  58) ;  the  general 
obliquity  of  the  rotation-vector  and  the  moment  of  momentum 
vector  is  one  characteristic  in  rotation,  and  is  operative  to  cause 
effects  to  which  there  is  no  parallel  where  a  kinematical  vector 
and  its  dynamical  associate  are  colinear,  Uke  momentum  and  its 
velocity.    If  angular  acceleration  is  absent,  everj^  element  in  (M') 


68  Fundamental  Equations  of  Dynamics 

is  zero,  but  (M")  is  not  affected,  since  it  depends  upon  the  (w) 
of  the  epoch,  and  not  upon  the  past  or  future  history  of  (ui) .  If  a 
rigid  body  is  spinning  steadily  about  a  fixed  axis  even,  (M")  is 
called  for,  as  a  directive  moment,  whenever  (o>)  and  (H)  diverge. 
For  the  case  of  rotation  about  the  center  of  mass,  (Mr")  will  be 
furnished  by  a  couple.  These  moments  are  recognizable  as  the 
centrifugal  couple  of  the  older  fashion  in  speech.  Like  forces 
normal  to  a  path,  they  disappear  from  the  power  equation  by  a 
condition  of  perpendicularity,  as  is  visible  from  equation  (68), 
when  we  have  noticed  through  equations  (75,  76)  that  (M")  is 
perpendicular  to  ((,i). 

56.  What  has  been  determined  about  (M")  presents  it  in  such 
relation  to  the  (o)  of  the  epoch  that  an  impressed  total  force- 
moment  of  that  value  is  adjusted  exactly  to  continuance  of 
constancy  in  the  rotation-vector  (o>);  the  zero  value  of  power 
and  the  consequent  constancy  of  (E)  being  an  evident  con- 
comitant of  that  as  primary  condition.  It  is  further  acceptable 
on  copimonsense  grounds  that  (H)  whose  divergence  from  (w) 
is  fixed  by  the  mass-distribution  when  (co)  is  constant,  as  the 
form  of  equation  (66)  proves,  must  then  accompany  that  mass- 
distribution  through  its  changes  in  azimuth  round  the  rotation- 
axis,  so  as  to  describe  a  right  circular  cone  and  keep  up  with  any 
originally  coincident  radius-vector  of  the  body.  And  the  shrink- 
ing of  such  a  cone  into  its  axis  provides  for  the  singular  case  of 
non-divergence,  with  no  (M")  required  for  adjustment. 

With  the  above  details  in  hand,  the  part  (M')  of  the  force- 
moment  appears  in  the  light  of  a  disturber  of  adjustment,  and 
that  opens  for  it  an  indefinite  range  of  possibilities  or  puts  away 
the  expectation  of  particular  conclusions,  except  two:  that  it 
must  supply,  first,  all  power  and  all  changes  in  magnitude  of  (H), 
and  secondly,  any  change  of  direction  that  displaces  (H)  rela- 
tively to  the  body. 

57.  At  this  point  the  chance  offers  for  a  pertinent  remark 


The  Fundamental  Equations  69 

about  all  equations  like  (74)  in  their  type.  They  exhibit  an 
impressed  phj'^sical  agency  (here  of  (M))  in  terms  that  compare 
it  for  excess  or  defect  with  an  adjustment  that  is  not  compensa- 
tion as  equilibrium  is,  but  calls  for  positive  action  (such  as  (M") 
here  exerts).  It  is  an  ambiguity  inseparable  from  the  algebra, 
especially  where  the  total  available  is  numerically  less  than  the 
critical  value,  that  an  adjustment  disturbed  is  indistinguishable 
from  one  not  secured.  In  other  words  we  can  be  sure  only  that 
(M')  and  (M")  are  mathematically  represented  in  (M),  when  the 
latter  has  been  assigned  arbitrarily;  using  again  the  present 
instance,  we  know  nothing  of  (M')  and  (M")  separately  as  active 
agencies.     Neither  of  the  forms 

M  =  M";        M  -  M"  =  0;  (77) 

indicates  equilibrium,  but  both  express  a  fulfilled  adjustment, 
much  as  equation  (36)  was  read.     Both  of  the  forms 

M  =  0;        M'  +  M"  =  0;  (78) 

apply  the  condition  of  equilibrium  to  (H)  in  the  sense  of  making 
it  a  constant  vector.  In  these  circumstances  an  angular  acceler- 
ation that  underlies  (M')  will  appear  in  the  equations  unless  (M') 
and  (M")  are  zero  separately,  which  can  be  true  only  specially; 
and  there  is  some  trace  of  mathematical  suggestion  that  this 
angular  acceleration  arises  by  give-and-take  between  (M')  and 
(M")  that  diverts  the  latter  from  its  original  office  of  keeping 
(fa))  constant. 

Doubtless  that  instinctive  view,  if  it  exists,  receives  some 
support  from  knowledge  of  other  conditions  in  which  an  active 
assignable  force-moment  is  indispensable  to  the  appearance  of 
angular  acceleration;  and  that  is  the  root  of  the  inclination  to 
see  paradox  in  the  phenomena  that  realize  the  conditions  of 
equation  (78).  But  in  consequence  of  the  divergence  already 
spoken  of,  if  the  (H)  vector  preserves  its  direction  in  the  reference- 


70  Fundamental  Equations  of  Dynamics 

frame  while  the  body  is  in  rotation,  the  vector  (w),  obHque  to  it, 
will  not  be  constant  also,  and  accordingly  there  will  be  angular 
acceleration.  This  occurs  spontaneously  we  might  say,  (M)  be- 
ing zero,  in  the  absence  of  control  that  would  be  effective  to 
keep  (o))  constant  and  shift  the  burden  of  change  upon  (H).  It 
makes  the  reasons  for  the  apparently  abnormal  results  more 
obscure,  that  the  kinematical  aspects  depending  upon  (<o)  and  (w) 
are  often  patently  visible,  whereas  the  dynamical  elements  that 
really  dominate  are  hidden  from  view.^ 

58.  While  we  are  laying  emphasis  upon  the  general  separation 
of  directions  for  (w)  and  (H),  it  is  proper  to  be  aware  how  this 
works  out  only  for  the  body  as  a  whole  through  the  mass-summa- 
tion of  (dH)  and  the  introduction  of  the  common  rotation-vector, 
and  does  not  appear  in  the  local  elements,  that  it  is  the  object  of 
that  plan  and  its  advantage  to  handle  in  one  group  and  not 
individually.  It  was  observed  already  in  equation  (2)  which 
had  not  yet  been  narrowed  to  rotation,  that  for  each  (dm)  its 
(dH)  and  its  (y)  are  coincident  vectors,  the  latter  lying  in  the 
normal  to  the  plane  (r,  ds)  and  being  attributed  to  the  local  (r)  as 
its  particular  angular  velocity.  This  lesson  can  now  be  repeated 
from  equation  (51)  or  (66),  if  we  denote  by  (wi,  Ti,  yi)  the  unit- 
vectors  of  (co)  and  of  (r),  and  of  the  perpendicular  to  (r)  in  the 
plane  (w,  r),  noticing  that  for  instance  equation  (66)  can  be 
written,  if  (a)  is  the  angle  (w,  r), 

dH  =  (o>i(ajr2)  —  Ti.{(ai^  cos  a))dm 

=  Yi(ajr^  sin  a) dm  =  ^(rMm).      (79) 

It  is  instructive  to  see,  next,  how  the  body  as  a  whole  retains 
for  its  total  moment  of  momentum  in  relation  to  its  rotation- 
vector  the  same  type  as  equation  (79)  shows;  and  this  can  be 
done  by  assembling  the  projections  of  every  (dH)  upon  the 
direction  of  (w).     The  result  to  be  recorded  for  use  is 

1  See  Note  18. 


The  Fundamental  Equations  71 

H(„)  =  /mO)i(cor2  -  (<oi-r)(G)-r))dm  =  wlu),  (80) 

expressed  as  we  find,  also  as  the  product  of  an  angular  velocity 
and  a  moment  of  inertia  about  its  axis,  but  both  these  factors 
now  refer  to  the  whole  body,  and  this  form  excludes  perpendicu- 
larity of  (<d)  and  (H). 

Because  (H)  is  a  sum  into  which  the  differently  weighted  ele- 
ments (y)  enter,  and  the  weighting  depends  upon  what  happens 
to  be  the  mass-distribution,  the  final  result  cannot  be  forced 
completely  into  any  one  mould,  beyond  the  point  here  estab- 
lished; only  we  know  that  the  rest  of  (H)  must  be  in  the  plane 
perpendicular  to  (<o).  Therefore  according  to  equation  (67)  we 
learn  that 

E  =  i("-H)  =f  lu),  (81) 

which  may  also  be  inferred  directly  from  equation  (52),  by  a 
slightly  varied  reduction  of  the  last  member  but  one.  Let  us 
use  the  occasion  to  renew  the  reminder  that  the  rotation  relative 
to  (C)  involves  only  a  transfer  to  its  notation  of  the  details  here 
attached  to  the  other  case. 

59.  A  similar  trend  can  be  marked  in  the  other  partners  (w) 
and  (M')  which  bring  kinematics  and  dynamics  into  connection : 
an  elementary  type  of  expression  which  appears  differentially 
then  persists  in  application  to  the  body  as  a  whole,  but  with  a 
supplement  governed  by  the  particular  mass-distribution  that 
produces  obliquity  of  (M')  and  (to).  For  the  local  element 
(dM')  equation  (74)  leads  by  expansion  to 

dM'  =  ((b(r-r)  -  r(a)-r))dm,  (82) 

which  it  will  be  noted  reproduces  equation  (66),  except  that  (w) 
has  replaced  (w)  throughout.  Consequently  equation  (79)  can 
be  paralleled  in  the  form 

dM'  =  (a)i(cbr2)  -  Ti(cor^  cos  j8))dm 

=  px(cor2  sin /S)dm  =  (w  son /3)pi(rMm).      (83) 


72  Fundamental  Equations  of  Dynamics 

But  (tbi,  ri,  pi)  are  now  unit-vectors  for  (to),  (r)  and  the  per- 
pendicular to  (r)  in  the  plane  (w,  r),  and  (j8)  denotes  the  angle 
(w,  r).  It  is  plain  that  (co  sin  j8)pi  is  for  each  (r)  the  effective 
part  of  (tb),  as  (w  sin  a)  yi  is  the  locally  effective  projection  of  (g>), 
and  that  (r^dm)  is  a  moment  of  inertia  for  the  axis  (pi).  Thus 
the  type  is  set  for  the  corresponding  expression  in  terms  devised 
to  apply  to  the  body;  and  in  fact  we  find 

M'(i)  =  /inw(r2  -  (<bi-r)2)dm  =  6l(^),  (84) 

whose  form  excludes  perpendicularity  likewise  for  (d>)  and  (M')- 

60.  It  can  be  conceded  as  one  legitimate  purpose  of  equations 
(80),  (81)  and  (84)  to  extract  from  the  more  general  treatment 
of  rotation  what  residue  of  correspondence  remains  with  those 
simpler  forms  that  are  met  in  uniplanar  dynamics.  Looking  in 
that  direction,  the  main  difference  can  be  localized  in  the  addi- 
tion of  an  independent  axis  of  (o)  to  stand  alongside  the  previous 
axis  of  (w).  But  the  greater  enlightenment  in  the  discussion 
comes  from  the  insistence  upon  putting  foremost  the  powerfully 
direct  analysis,  by  means  of  the  dynamical  vectors  (H)  and  (M) 
and  their  connections.  This  tends  to  make  the  kinematical 
vectors,  and  especially  (&>),  rather  subsidiary  until  restrictions 
upon  the  problem  restore  to  them  more  nearly  equal  weight. 

61.  If  we  start  again  from  equation  (66)  and  enter  upon  the 
semi-cartesian  expansion  for  the  vector  (H)  the  first  results  found 
are 

H(x)  =  co(x)/m(r-r)dm  -  /mX(<o-r)dm;  1 

H(y)  =  fa>(y)/m(r-r)dm  -  /n,y(o)-r)dm;  [■  (85) 

H(z)  =  fc)(z)/m(r-r)dm  -  /„iZ(fa>-r)dm.  J 

These  continue  to  assume  pure  rotation  round  the  origin,  the 
body  being  in  a  general  orientation  relative  to  the  reference- 
frame  (XYZ).  Retaining  one  value  of  (o>)  given  in  relation  to 
(XYZ),  the  last  terms  in  the  second  members  are  seen  to  depend 
upon  the  body's  orientation,  but  the  first  terms  are  invariant  for 


The  Fundamental  Eguaticyis  73 

all  such  orientations.  By  a  definite  choice  of  orientation  the  last 
terms  can  always  be  remarkably  simpUfied,  and  what  are  known 
as  the  principal  axes  of  inertia  for  the  origin  will  then  coincide 
with  the  axes  (XYZ).  We  presuppose  the  proof  that  there  are 
never  fewer  than  three  orthogonal  principal  axes  at  every  point 
that  is  in  rigid  configuration  with  a  rigid  body,  and  ordinary 
acquaintance  with  properties  of  the  ellipsoid  of  inertia  or  mo- 
mental  ellipsoid;  this  material  is  standard  and  accessible. 

In  all  three  equations  expand  (to-r)  and  reduce  to  the  forms 

H(x)  =  i{co(x)I(x)  —  co(y)/mXydm  —  co(2)/mZxdm};    1 

H(y)  =  j{co(y)I(j)  -  aj(,)/myzdm  -  oj(x)/mXydm} ;    V      (86) 

H(j)   =  k{aj(z)I(2)  —  a)(x)/razxdm  —  co(y)/myzdm}.    J 

The  property  of  principal  axes  determines  the  disappearance  of 
six  integrals  at  the  orientation  where  those  lines  of  the  body 
coincide  with  (XYZ).  Supposing  that  coincidence,  therefore,  it 
becomes  true  that 

H  =  w(x)I(x)  +  <«>(y)I(y)  +  w(z)I(i)-     [Principal  axes.]     (87) 

But  (H)  can  be  represented  invariantly  by  an  indefinite  number 
of  groups  of  orthogonal  projections,  and  for  one  group,  which 
can  be  chosen  at  every  epoch  and  for  every  (w),  the  coincidences 
that  simpUfy  equation  (87)  will  occur  instantaneously.  How 
and  on  what  terms  the  advantage  of  the  simplification  can  be 
made  permanently  available  is  a  question  to  be  taken  up  here- 
after (see  section  118);  but  some  useful  decisions  follow  immedi- 
ately here. 

62.  And  first,  the  possible  extent  is  made  evident  of  the  can- 
cellation ensuing  through  the  difference  between  the  two  con- 
tributions to  the  second  member  of  equation  (66) .  It  is  indicated 
by  the  present  remainder,  in  which  all  the  terms  are  essentially 
positive,  if  we  take  the  vector  factors  absolutely.  Secondly,  if 
we  turn  to  kinetic  energy,  the  aid  given  by  adopting  principal 


74  Fundamental  Equations  of  Dynamics 

axes,  there  too,  is  apparent  in  reducing  the  number  of  terms  in 
the  expression.  For  whereas  the  expansion  of  equation  (67)  on 
the  basis  of  equation  (86)  will  yield  nine  terms  that  do  not 
coalesce  into  fewer  than  six,  the  reduction  of  these  to  three  is  a 
consequence  of  equation  (87),  from  which  follows 

E    =   ■|[(w(x))^I(x)    +   (W(y))''l(y)    +   (aJ(^))2I(,)]. 

[Principal  axes.]      (88) 

This  again  by  deleting  subtractive  terms  has  regained  parallelism 
with  the  case  of  translation  and  three  orthogonal  components  of 
velocity  except  for  the  difference,  irreducible  in  the  general 
expression,  between  the  uniform  mass-factor  (m)  and  the  indi- 
vidual inertia-coefficients  Hke  (I(x)). 

63.  In  the  third  place,  that  similarity  in  type  between  equa- 
tion (66)  and  equation  (82)  which  has  been  relied  upon  before  to 
abbreviate  details  can  be  employed  again.  Like  equations  (86) 
for  (H)  we  can  write  for  (M') 

M'(x)  =  i{w(x)I(x)  —  a>(y)/mxydm  —  W(z)/mzxdm};  "1 

M'(y)    =  j{w(y)I(y)    —    W(z)/myzdm  —   W(x)/niXydm } )    [■      (89) 

M'(z)  =  k{co(z)I(z)  —  co(x)/mZxdm  —  w(y)/myzdm};  J 

in  which  the  same  six  integrals  occur  that  the  choice  of  principal 
axes  eliminates.  Consequently  if  we  use  at  the  epoch  the  pro- 
jections upon  the  principal  axes,  we  obtain 

M'  =  W(x)I(x)  +  w(y)I(y)  +  w(z)I(z).     [Principal  axes.]       (90) 

This  adds  one  feature  to  the  previous  conclusion  in  equation 
(84),  and  makes  evident  that  (M')  cannot  vanish  while  (w)  differs 
from  zero,  as  a  limitation  upon  the  subtractive  element  of  equation 
(82) .  And  it  throws  stronger  light  upon  a  possible  constancy  of  (E) 
while  both  (M')  and  (M")  are  active,  for  which  the  condition  is 
that  (M')  as  well  as  (M")  should  be  perpendicular  to  (w).  This 
is  compatible  with  the  presence  of  («)  since  the  latter  may  have 
any  direction  relatively  to  (w).     Where  the  time-derivatives  of 


The  Fundamental  Equations  75 

equations  like  (80)  play  a  part  in  such  considerations  as  the  fore- 
going, of  course  it  may  be  necessary  to  take  account  of  variable 
moment  of  inertia  as  being  important  in  reconcihng  the  presence 
of  (ti)  and  the  absence  of  (M). 

Should  the  rotation  that  is  under  investigation  be  about  the 
center  of  mass  of  the  body,  the  force  to  be  brought  in  for  the 
accompanying  translation  or  to  accelerate  the  particle  of  the 
combination  is  calculable  as  (m^),  where  any  value  may  have 
been  assigned  by  other  elements  to  the  second  factor.  But  if  the 
case  is  one  of  pure  rotation  round  any  origin  or  fixed  point,  it  is 
plain  that  the  acceleration  and  velocity  of  the  center  of  mass  are 
prescribed  at  the  values 

t  =  (g)  X  f)  +  (w  X  v);        V  =  (fc)  X  f),  (91) 

requisite  locally  under  the  rule  of  equations  (44,  72).  Then  the 
total  force  brought  to  bear  must  be  accurately  adjusted  to  produce 
this  acceleration,  and  a  constraint  at  the  origin  may  have  to  be 
made  active  in  order  to  give  exactly  the  requisite  force.  For 
reasons  of  that  nature,  the  constraint  may  need  to  be  calculated 
or  expressed,  although  it  can  contribute  nothing  to  the  moment 
(M)  about  the  origin,  and  can  in  that  respect  be  ignored.  It 
rests  upon  the  general  understanding  about  sections  45  and  51, 
that  all  the  leading  equations  like  (86,  88,  89)  are  adaptable  to 
center  of  mass  as  origin  without  formal  change,  and  by  mere 
substitution  of  the  values  then  effective. 


CHAPTER   III 
Reference-Frames:  Transfer  and  Invariant  Shift 

64.  Let  us  recall  now  the  fact  that  the  exercise  of  choice  of 
reference-frame  must  be  an  assumed  preliminary  to  determining 
any  definite  working  values  for  the  fundamental  quantities,  and 
consequently  for  all  quantities  calculable  in  terms  of  them. 
This  is  not  interfered  with  as  a  truth  by  our  predominant  habit 
of  making  the  earth's  surface  locally  the  tacitly  adopted  basis  of 
reference.  The  circumstances  then  bring  with  them  quite  natu- 
rally a  recognizable  need  of  deliberately  guided  inquiry  into  the 
extent  to  which  such  values  are  affected  by  an  allotted  range  in 
selection  and  specification  for  our  reference-frame.  This  will 
afford  the  necessary  machinery  for  correct  transfer  from  one 
reference-frame  to  another  as  standard  when  that  is  dictated  by 
an  effort  at  greater  precision  or  by  reasons  founded  in  an  ad- 
vantage of  convenience. 

The  line  of  thought  to  be  taken  up  next  will  trace  out  those 
matters  of  material  consequence  connected  with  the  chief  kine- 
matical  and  dynamical  expressions  which  require  for  their  settle- 
ment a  collation  of  values  resulting  when  particular  frames  are 
chosen  among  a  group  that  are  in  assigned  conditions  of  relative 
configuration  and  motion.  The  fullest  survey  belonging  to  that 
discussion  embraces  much  that  would  be  scarcely  relevant  on 
the  scale  laid  down  for  our  present  undertaking.  But  by  allowing 
the  more  practical  interests  in  these  directions  to  set  the  limits, 
we  shall  confine  our  scope  to  methods  that  are  in  most  frequent 
use  for  translating  the  important  expressions  into  convertible 
terms  of  familiar  type  and  ascertaining  their  mutual  dependence. 
In  so  far  as  vectors  can  be  made  the  vehicle  of  expression,  they 

76 


Reference  Frames  77 

are  likely  to  deal  direct!}^  with  resultants  and  totals,  and  then 
we  are  concerned  with  the  amounts  by  which  these  change  at  a 
transfer  from  one  frame  to  another.  Yet  because  we  must  at 
times  prepare  more  completely  for  computation,  this  alone  would 
constrain  us  to  sacrifice  to  those  ends  the  compactness  of  vectorial 
statements.  Other  reasons  also  compel  us  to  find  place  for  the 
partials  or  components  that  are  characteristic  of  various  coordi- 
nate systems  whose  peculiar  advantages  make  them  useful 
auxiliaries  to  the  reference-frame;  and  this  will  raise  a  second 
group  of  questions.  Some  close  intrinsic  connections  will  be 
found,  however,  to  make  interdependent  the  two  branches  of  the 
inquiry,  relating  one  to  the  uses  of  coordinate  systems  and  the 
other  to  comparisons  among  reference-frames,  which  occupy  this 
chapter  and  the  next. 

65.  First  as  to  transfers  and  comparisons  among  reference- 
frames.  Since  scalar  mass  that  is  unaffected  by  position  and 
motion  becomes  by  that  supposition  neutral  to  the  main  issues 
here,  something  can  be  done  toward  clearing  the  ground  by 
noticing  at  once  how  many  important  decisions  must  then  turn 
upon  the  kinematical  factors;  solely  upon  these  in  the  differ- 
ential elements,  though  as  we  have  found  at  certain  points  in 
the  preceding  chapter,  the  mass-distribution  continues  to  play 
some  part  through  the  integrals  that  are  related  to  the  center  of 
mass  and  to  the  moments  of  inertia.  Accordingly  we  are  enabled 
to  restrict  ourselves  in  the  first  steps  to  kinematics,  essentially 
to  radius-vectors  and  velocities  and  accelerations,  the  properly 
dynamical  phase  being  covered  finally  by  introducing  the  neces- 
sary mass  factors. 

As  one  aid  to  brevity,  we  shall  outline  a  notation  by  way  of 
preface,  to  be  used  consistently  throughout  the  combinations  and 
comparisons  that  we  must  make.  Let  one  reference-frame  estab- 
lished by  its  origin  (O)  and  its  axes  (XYZ)  be  constituted  the 
standard,  the  axes  being  orthogonal  and  in  the  cycle  of  a  right- 


78  Fundamental  Equations  of  Dynamics 

handed  screw.  By  affording  to  our  thought  one  term  common 
to  a  series  of  comparisons,  this  frame  will  furnish  a  means  of 
coordinating  their  individual  results.  Let  any  one  of  the  other 
reference-frames  with  which  we  may  happen  to  be  concerned 
alternatively,  either  under  suggestion  from  special  conditions  or 
for  the  purpose  of  more  general  discussion,  be  determined 
through  its  origin  (O')  and  its  axes  (X'Y'Z')  and  be  distinguished 
as  a  comparison-frame.  All  the  frames  are  supposed  congruent. 
We  shall  preserve  a  helpful  symmetry  of  notation  by  assigning 
regularly  primed  quantities  to  comparison-frames  and  unaccented 
symbols  to  the  standard.  But  we  must  not  fail  to  remember 
either  that  the  distinction  which  sets  off  one  frame  as  standard 
is  for  convenience  of  correlation  only,  in  the  first  instance,  and 
it  retains  its  arbitrary  element  until  physical  reasoning  can  be 
seen  to  converge  noticeably  or  convincingly  upon  one  frame,  or  a 
set  of  frames  meeting  formulated  conditions,  as  the  basis  better 
accommodated  to  the  ultimate  statement  of  any  physical  laws 
or  regular  sequences  among  phenomena.  We  have  touched  on 
this  point  in  sections  6  and  7.  In  the  preliminary  view  every 
frame  is  qualified  for  selection  to  be  standard,  in  relation  to 
which  all  the  others  fall  into  their  status  of  comparison-frames. 
66.  The  configuration  of  any  (0',  X'Y'Z')  relative  to  the  stand- 
ard can  be  specified  as  though  it  had  arisen  in  virtue  of  a  dis- 
placement from  original  coincidence  with  (O,  XYZ),  without 
needing  to  imply,  however,  that  the  coincidence  once  existed  in 
reality  and  that  the  final  configuration  has  developed  pro- 
gressively by  a  time  process,  but  also  without  excluding  the 
latter  possibility.  In  order  to  dispose  of  certain  aspects  of  the 
matter,  let  us  at  first  conceive  definitely  all  these  individual 
configurations  to  be  permanent,  each  comparison-frame  being 
taken  in  a  configuration  that  it  retains.  Then  any  continuous 
transitions  within  an  arrangement  of  such  frames  will  associate 
themselves  rather  with  grouping  it  into  a  space  locus,  and  no 


Reference  Frames  79 

idea  will  be  imported  into  it  of  those  other  features  belonging 
distinctively  to  motion  and  a  path.  But  we  must  expect  to  find 
here  as  elsewhere,  that  the  two  points  of  view  run  easily  one  into 
the  other,  with  those  groups  of  virtual  displacements,  indicated 
as  possible  without  violating  the  conditions  for  the  locuS;  becom- 
ing an  actual  series  in  time  when  the  paths  are  described.  One 
moving  frame  can  mark  the  positions  of  all  members  of  a  group 
that  are  in  permanent  configurations,  as  it  coincides  with  them 
in  succession.  In  point  of  fact,  several  similar  modulations  of  the 
thought  here  hinge  alike  upon  that  dual  conception  of  the 
elements  that  enter. 

67.  The  assignment  of  its  relative  configuration  will  involve  in 
general  for  any  frame  both  a  difference  of  position  between  (0') 
and  (O)  and  a  difference  of  orientation  between  (X'Y'Z')  and 
(XYZ).  Moreover  these  two  data  are  assignable  independently, 
and  it  is  intuitively  true  that  the  actual  localization  of  (O', 
X'Y'Z')  is  reproducible  from  coincidence  with  (O,  XYZ)  by  com- 
bining them  in  either  order.  Let  the  parallel  displacement  or 
translation  of  the  axes  with  the  origin  (0')  be  specified  by  the 
vector  (00')  which  we  shall  denote  by  (ro).  And  the  changed 
orientation  is  equivalent  to  a  subsequent  displacement  by  rota- 
tion of  (X'Y'Z')  as  a  rigid  cross,  because  they  are  congruent  with 
(XYZ)  and  remain  orthogonal.  Using  the  notation  of  section  45, 
we  can  indicate  the  result  by  the  vector  sum 

Y  =  /dr,  •    (92) 

with  the  possibility  attaching  to  resultants  in  general,  of  repre- 
senting equivalently  many  sets  of  components. 

If  the  idea  of  succession  enters  the  last  equation,  the  present 
connection  confines  it  to  a  timeless  series  of  elements  (dy),  in 
each  of  which  the  constituents  (^cyv)  or  substitutes  for  them  are 
coexistent.  Where  it  will  not  cause  confusion,  the  term  rotation- 
vector  can  be  applied  to  (dy),  as  well  as  to  (y)  of  the  earlier 


80  Fundamental  Equations  of  Dynamics 

section.  For  any  comparison-frame  accordingly  its  configuration 
is  given  with  the  requisite  definiteness  by  the  two  total  displace- 
ments taken  in  either  order, 

To  =  /dio;        Y  =  fdy.  (93) 

68.  Let  us  introduce  next  any  point  (Q)  having  at  a  given 
epoch  radius-vector  (r)  in  the"  standard,  and  (r')  in  a  comparison- 
frame.  The  difference  of  orientation  alone  while  (0')  coincided 
with  (O)  would  leave  the  radius-vector  invariant  for  all  per- 
missible sets  of  axes,  the  expression  of  which  condition  can  be 
put  into  terms  of  the  two  sets  of  unit-vectors, 

r  =  ix  +  jy  +  kz  =  i'x'  +  j'y'  +  k'z';  (94) 

where  the  invariance  is  noticeably  obscured  until  the  vector 
algebra  brings  it  into  full  rehef.  The  alternative  relation 
accompanying  separation  of  (0')  and  (O)  is 

r  =  To  +  r',  (95) 

whose  form  obviously  excludes  equality  of  (r)  and  (r')  so  long 
as  (ro)  differs  from  zero.  It  should  be  observed  about  the  last 
equation  that  it  is  based  rather  upon  a  triangle  as  graph  than 
upon  a  parallelogram,  because  the  conception  of  (r')  makes  it  a 
localized  vector  with  (O')  for  base-point. 

Regarding  now  (Q)  as  typical  in  any  continuous  or  discon- 
tinuous assemblage  of  points,  and  (Q')  as  any  other  such  point 
whose  radius-vectors  in  the  two  frames  appear  in  the  allowable 
forms  (r  +  Ar),  (r'  +  Ar'),  we  have  for  the  vector  (QQ') 

Ar  =  Ar',  .  (96) 

throughout  the  group  of  points,  independently  of  the  points 
chosen  and  of  the  particular  comparison-frame  employed.  This 
records  the  patent  truth  that  the  arrangement  of  members  in 
any  point-group,  or  their  configuration  relative  to  each  other,-  is 
expressible  invariantly  by  means  of  the  standard  frame  and  of 


Reference  Frames  81 

every  (O',  X'Y'Z').  With  that  meaning  the  remark  is  to  be 
accepted  that  "  Position  coordinates  appear  in  our  equations  by 
a  convenient  fiction  only,  they  being  parasitic  and  auxihary 
variables  that  can  be  eliminated."^ 

69.  If  for  sufficient  reason  we  maintain  the  discrimination 
between  (Q)  and  (Q')  as  two  individual  points  and  locate  each 
permanently  in  its  configuration  with  (O,  XYZ) ;  or  let  each  be 
fixed  in  the  space  attached  to  the  standard  reference-frame  in  the 
words  of  one  current  phrase;  no  questions  about  time-derivatives 
of  (r),  (r'),  (Ar)  or  (Ar')  can  arise,  so  long  as  the  configuration  of 
(O',  X'Y'Z')  is  also  by  supposition  permanent.  The  source  of 
those  reasons  and  their  cogency  will  depend  upon  the  case  in 
hand;  they  may  be  physical  in  their  nature  and  extracted  by 
interpretation  and  analysis  from  observation,  or  their  origin^ 
may  be  frankly  due  to  a  feature  in  the  mathematical  treatment. 
By  associating  other  such  individual  points  with  (Q)  and  (Q') 
we  may  build  up  a  continuous  group  as  a  limit,  for  which  the 
general  radius-vector  becomes  in  length  a  function  of  its  orienta- 
tion but  the  essentials  of  the  description  remain  timeless. 

However  in  any  unforced  survey  of  other  particular  circum- 
stances and  their  plain  suggestions  a  competitive  view  must  find 
recognition,  that  will  regard  both  (Q)  and  (O',  X'Y'Z')  as  indi- 
viduals somehow  identifiable  through  a  series  of  changing  con- 
figurations in  (0,  XYZ),  and  consequently  any  account  that  aims 
at  practical  completeness  cannot  neglect  coordinating  the  two 
alternatives.  There  is  the  elementary  fact,  for  example,  that  the 
same  dependence  of  radius- vector  upon  its  orienting  angle  as  before 
can  be  presented  with  both  variables  made  functions  of  time.  But 
the  fruits  of  that  idea  are  not  exhausted  in  one  announcement  at 
the  threshold  of  the  matter.  For  when  in  our  view  (Q')  becomes 
a  subsequent  position  of  the  point  (Q),  or  whenever,  more  inclu- 
sively, the  varying  position  of  a  moving  point  is  matched  at  each 

1  Quoted  from  Poincare. 


82  Fundamental  Equations  of  Dynamics 

epoch  with  the  permanent  position  of  a  coincident  point,  the 
twofold  relation  of  the  same  symbols  to  which  this  leads  with 
such  a  double  point  will  reappear  perpetually.  This  can  make 
either  aspect  of  the  coincidence  a  continuous  indicator  or  marker 
for  the  other,  by  means  of  some  connecting  rule  that  formulates 
from  either  side  the  relation  of  consecutive  values — here  of  the 
radius-vector.  Neither  phase  of  the  combination  can  be  ignored 
or  subordinated,  without  losing  hold  upon  ideas  that  are  central 
in  evaluating  any  variable  quantity  by  legitimate  transition  to  a 
substituted  uniform  condition.^ 

70.  These  considerations  confront  us  with  the  necessity  of 
preparing  here  for  that  kind  of  transition,  and  conceiving  (Q) 
and  (0',  X'Y'Z')  to  be  individual  and  moving.  This  can  be 
executed  conveniently  by  subdividing  into  steps,  and  taking  first 
the  one  that  affects  (Q)  alone,  while  we  retain  for  the  time  being 
that  permanent  configuration  of  (0',  X'Y'Z')  in  the  standard 
frame  which  is  afterwards  to  be  abandoned.  If  we  accept  for 
(Q)  and  (Q')  a  fusion  of  identity  in  the  sense  that  they  are  now 
adopted  as  two  positions  of  the  same  moving  point,  terminal  for 
any  time-interval  (At),  the  mean  velocities  for  that  interval  will 
be  equal  in  our  two  reference-frames,  and  also  the  instantaneous 
velocities  at  the  epoch  beginning  the  interval.  This  conclusion 
finds  expression  in  sequence  with  the  requisite  new  reading  of 
equation  (96)  as 

V  =  Lim^t=o  (  ^  )  =  Lim^t=o  (~^]-^''>  (^'^) 

or  in  semi-cartesian  dress, 

dx       .  dy      ,  dz       ..dx'       .,dy'  .   ,,dz'        ,      ,„^, 

V  =  iTr  + j^f  +  k^TT  =  i'-rr-f  j'-57 +k'-r-  =  v',     (98) 

dt       Mt  dt  dt       •*    dt  dt  '     ^     ^ 

since  both  sets  of  unit-vectors  are  by  supposition  constant  here, 
as  well  as  (lo).     And  further,,  because  these  simultaneous  veloci- 
1  See  Note  19. 


Reference  Frames  83 

ties  of  (Q)  are  thus  continually  equal  vectors,  it  is  an  evident 
corollary  that  the  accelerations  of  (Q)  in  the  two  frames  are 
always  equal  at  the  same  epoch;  or 


t  -  Lim,t=o(|^)  =  Lim,t=o(^)  -  t', 


(99) 


whose  expanded  equivalent  again  is 

.  d^x       .  dV      ,  d-z 
^  ^  ^dt^  +  ^dt^""^     dt2 

.,dV      .dy      ,,dV        ,      ,      , 
dt2  ^  ^   dt2  ^      dt2  ^^""^ 

71.  Taken  together,  these  statements  make  clear  for  every 
epoch  the  in  variance  of  velocity  and  acceleration  that  holds 
good  throughout  any  group  of  reference-frames  that  are  in 
permanent  relative  configuration.  Also  the  consequences  in 
application  to  the  same  system  of  bodies  at  the  same  epoch  are 
apparent.  Each  local  velocity  and  acceleration  being  un- 
affected, the  six  fundamental  quantities  show  in  the  standard  and 
in  any  comparison-frame  of  the  group  thus  correlated: 

Q  =  q',        R  =  R';        E  =  E';        P  =  P';  1 

H  =  H'  -F  (ro  X  QO;        M  =  M'  +  (ro  x  R'),  J      ^     ^ 

which  it  may  be  well  to  compare  for  likeness  and  difference,  say 
when  (ro  =  f),  with  the  corresponding  relations  exhibited  in 
section  51,  the  contrast  between  (C)  there  and  (O')  here  lying  in 
the  freedom  of  the  former  point  to  move  with  velocity  and  ac- 
celeration. The  less  narrowly  limited  connection  of  center  of 
mass  with  force-moment  and  moment  of  momentum  should  be 
realized. 

72.  The  foregoing  results  are  sufficiently  practical  in  their 
bearing  to  incite  us  to  appropriate,  without  delaying,  the  possi- 
bilities that  they  illustrate.  These  lie  in  the  direction  of  a  certain 
liberty  to  employ  what  amounts  to  a  whole  series  of  different 


84  Fundamental  Equations  of  Dynamics 

reference-frames  at  successive  epochs  of  the  same  problem,  or 
inside  the  range  covered  by  one  discussion,  and  yet  avoid  pro- 
hibitive compHcations  that  might  be  due  to  such  repeated  trans- 
fers to  new  standards.  Provided  only  that  we  observe  those 
restrictions  which  underlie  the  invariance  of  any  particular 
quantities  with  which  we  are  dealing,  the  frames  become  inter- 
changeable in  respect  to  them;  and  freedom  prevails  to  depart, 
at  later  epochs  and  as  often  as  may  prove  desirable,  from  the 
initial  choice  of  reference-frame.  At  least  it  is  evident  how 
there  will  be  no  danger,  on  relinquishing  one  frame  and  adopting 
another  subject  to  the  proper  conditions,  of  dislocating  ruinously 
by  breaking  into  it  the  expression  of  a  continuous  series  of  values 
for  any  quantity  that  the  change  leaves  invariant.  Dislocations 
of  minor  scope  can  be  reckoned  with  otherwise,  or  often  dis- 
regarded, where  they  enter. 

Such  procedure  remains  clearly  valid,  always  within  its  limita- 
tions, whether  its  revisions  of  choice  involve  configurations 
separated  by  steps  that  are  finite  or  that  are  made  with  finite 
pauses  between  them,  or  whether  the  group  of  frames  used  melts 
at  the  limit  into  a  continuously  consecutive  arrangement.  It  is 
equally  permissible,  besides,  to  regulate  the  employment  of 
members  in  a  group  of  frames  according  to  a  time-schedule,  or 
to  effect  timeless  transitions  among  frames  and  to  concern  our- 
selves comparatively  with  simultaneous  values  of  different  quan- 
tities, or  finally  of  the  same  quantity  when  we  break  the  barrier 
of  invariance.  The  actual  working  out  of  the  main  thought 
rings  the  changes  on  all  these  offered  chances,  so  that  several  of 
the  combinations  will  come  before  us  prominently  for  specific 
examination. 

73.  We  proceed  next  to  remove  the  limitation  that  has  held 
us  to  permanent  configuration  for  (O',  X'Y'Z').  We  relax  this 
permanence  relative  to  (O,  XYZ)  by  admitting,  first  changes  in 
(ro)  alone  while  (y)  is  unchanging  in  equation  (93),  and  after- 


Reference  Frames  85 

wards  the  full  freedom  with  changes  in  (y)  also.  It  seems  ad- 
vantageous to  attack  this  phase  of  the  matter,  too,  through  what 
we  have  spoken  of  as  fusion  of  identity;  but  now  for  comparison- 
frames  that  like  the  points  (Q)  and  (Q')  can  from  another 
approach  also  be  distinguished  as  separate  individuals.  Return 
then  to  that  original  view  of  those  points,  include  some  second 
comparison-frame  (O",  X"Y"Z")  and  carry  on  the  notation  by 
adding 

O'O"  =  Aro;        0"Q'  =  r".  (102) 

The  relations  associating  (Q)  with  (0')  and  (O),  and  (Q')  with 
(0")  and  (O)  are 

r  =  To  +  r';        r  -j-  Ar  =  (ro  +  Aro)  +  r",  (103) 

showing  by  their  difference 

At  =  Ato  +  (r"  -  r'),  (104) 

whose  verbal  equivalent  can  be  read  from  the  broken  line 
,(QO'0"Q')  that  is  equal  as  a  vector  sum  to  (QQO  and  closes  a 
quadrilateral  that  may  be  of  course  either  gauche  or  plane. 

We  may  now  retrace  the  previous  track  further,  whenever  we 
can  attribute  to  the  frames  (0^  X'Y'Z')  and  (O",  X"Y"Z") 
some  adequate  basis  of  continuous  identity  similar  to  that  which 
was  made  to  unite  (Q)  and  (Q'),  so  that  the  entire  group  of 
discrete  frames  of  permanent  but  differing  configurations  is 
replaced  by  the  conception  of  one  representative  frame  (O', 
X'Y'Z')  in  continuously  variable  relation  to  the  standard.  First, 
confine  attention  to  the  origin  (O'),  deferring  a  little  the  intro- 
duction of  changing  orientation,  suppose  (ro)  to  vary  with  time 
and  read  equation  (104)  to  correspond.  The  originally  un- 
related vectors  (r')  and  (r")  coalesce  under  one  symbol  (r') 
when  that  is  used  to  signify  a  vector  drawn  always  from  the 
position  of  (O')  at  any  epoch  to  the  simultaneous  position  of  (Q). 
It  is  therefore  a  vector  to  be  rated  in  the  standard  frame  as 


86  Fundamental  Equations  of  Dynamics 

localized,  but  variable  in  all  three  particulars  of  length,  orienta- 
tion and  base-point.     In  pursuance  of  that  thought  write 

r"  -  r'  =  Ar',  (105) 

divide  equation  (104)  by  the  elapsed  time  (At)  and  proceed  to 
record  the  limiting  ratio  in  the  form 


(S) 


Lim^t=a  ['^^J  =  r  -  f  0  =  V  -  Vo,  (106) 

if  (v)  and  (vo)  denote  the  velocities  of  (Q)  and  of  (0')  in  the 
standard  frame.  The  formal  repetition  in  this  first  member  of 
(v')  as  specified  in  the  terms  of  equation  (97)  is  significant  of  its 
unconstrained  meaning  here  too  as  the  velocity  of  (Q)  reckoned 
in  the  frame  (O',  X'Y'Z'),  but  under  an  extension  that  allows  a 
supposed  motion  of  (O')-  Duly  observing  the  imposed  condition 
of  unchanging  orientation  for  (i'j'k')  that  is  still  maintained, 
confirm  this  feature  of  the  development  by  writing  the  time- 
derivative  of  the  permanent  relation  in  equation  (95)  in  the  form 

..  dx'       .,  dy'       ,  ,  dz'  ,     _, 

and  compare  with  equation  (98).  It  is  plain  that  (v')  and  (v.) 
are  equal  at  any  epoch  when  (f  o)  is  zero. 

74.  These  thoughts  harmonize  in  another  respect  with  equa- 
tion (106)  if  we  see  registered  there  a  consequence  of  a  double 
process  of  incrementation  for  the  vector  (r'),  now  completely  vari- 
able in  the  standard  frame,  with  rate  (v)  at  its  forward  end  and 
with  rate  (vo)  at  its  base-point.  In  every  such  combination,  so 
long  as  these  rates  are  equal,  the  vector  retains  its  length  and 
orientation  in  the  reference-frame;  as  a  free  vector  it  remains 
equal  at  all  epochs,  though  as  a  localized  vector  it  experiences 
change  of  position  determined  by  the  common  value  of  the  two 
rates.  In  the  less  particularly  chosen  suppositions  where  the  two 
rates  are  unequal,  only  their  difference  such  as  (v  —  Vo)  is  avail- 
able to  give  change  of  tensor  and  of  orientation. 


Reference  Frames  87 

But  to  take  account  of  these  latter  elements  for  (r')  and  to 
ignore  or  drop  out  the  change  in  position  for  (O')  substitutes 
effectively  (O',  X'Y'Z')  as  reference-frame,  the  orientation  of 
(i'j'k')  having  first  and  last  the  requisite  permanence,  so  that  the 
transfer  is  uncomphcated  in  that  respect.  And  since  the  part 
(vo)  applies  simultaneously  or  in  common  to  all  points  (Q),  the 
readjustment  of  velocity  values  made  necessary  by  this  type  of 
transfer  to  a  new  reference-frame  (O',  X'Y'Z')  can  be  summarized 
as  the  subtraction  of  a  translation  with  the  velocity  of  the  new  origin 
in  the  first  standard  frame.  In  connection  with  this  the  thought 
frequently  finds  expression  that  each  frame  carries  its  space  in 
rigid  attachment  to  it,  and  these  interpenetrating  spaces  will 
have  in  the  present  case  at  each  coincident  pair  of  points  the 
relative  velocity  (±  Vo)  at  any  epoch. 

The  effects  upon  acceleration  of  a  similar  transfer  while 
(i'j'k')  remain  constant  show  plainly  on  forming  the  time-deriva- 
tive of  equation  (107).     This  gives 

d-x'  d-v'  d^z' 

t'^i'-^-+j'-^^  +  k'— =  v- Vo;    ^t-t'  =  vo;     (108) 

and  the  proper  allowance  shows  again  in  terms  of  a  translation 
with  the  new  origin  (O'),  whose  acceleration,  however,  is  now 
essential  and  not  its  velocity.  In  the  hght  of  equations  (107, 
108)  the  combinations  become  self-evident  by  which  velocities 
or  accelerations  or  both  may  be  left  invariant  under  a  change  of 
reference-frame.  The  bearing  upon  the  segregation  in  sections 
21,  31,  48  and  49  will  not  escape  attention. 

75.  In  order  to  embrace  finally  the  transition  to  axes  (X'Y'Z') 
whose  orientation  is  changing  in  the  standard  frame,  while  they 
are  accompanying  their  origin  (O'),  we  can  use  our  knowledge 
that  the  rotation-vector  of  sections  45  and  67  specifies  such 
changes  adequately,  and  thus  complete  under  the  wider  play  of 
these  conditions  the  time-derivative  of  the  relation  that  remains 
valid, 


88  Fundamental  Equations  of  Dynamics 

r  =  ro  +  r'  =  To  +  (i'x'  +  j'y'  +  k'z').  (109) 

Upon  the  supposition  that  the  group  (i'j'k')  are  at  the  epoch 
varying  in  direction  relative  to  (XYZ)  as  determined  by  the 
rotation- vector  (y),  we  are  led  by  the  differentiation  directly  to 
the  equation 

'  =  '»+(txr')  +  (4f  +  j'f +  k'f),    (110) 

from  which  it  follows  that 

V- v' =  Vo  + (i-xrO;        v  =  v' +  [vo  +  (y  x  r')].    (HI) 

Typical  special  cases  under  this  equation  can  be  decided  by 
inspection.  Note  the  form  now  taken  by  the  idea  of  inter- 
penetrating spaces  in  section  74,  connecting  it  with  the  general 
motion  of  a  rigid  solid  in  section  48.  The  last  group  of  terms  in 
equation  (110)  must  still  be  recognized  as  the  velocity  (v')  of 
(Q)  in  (O',  X'Y'Z'),  because  the  transfer  to  the  latter  as  the 
standard  cancels  perforce  from  admission  into  (v')  every  change 
in  orientation  attributable  otherwise  to  (i'j'k'),  in  addition  to 
ignoring  changes  in  the  position  of  (O'). 

76.  Various  equivalent  verbal  formulations  beside  those  al- 
ready suggested  can  be  devised  for  equations  (107,  108,  111), 
that  all  amount  in  principle  to  a  superposition  of  relative  veloci- 
ties or  accelerations.  And  it  will  be  seen  how  the  same  idea  can 
be  applied  repeatedly  and  can  carry  us  through  a  chain  of  trans- 
fers to  a  final  result  that  accumulates  in  itself  all  the  contributions 
at  its  several  steps.  Remembering  that  forces  are  bound  to 
superposition  also,  as  they  enter  successively  with  the  acceptance 
of  their  accelerations  into  physical  status,  trace  there  a  line  of 
advance  in  precision  that  would  parallel  our  discarding  one 
reference-frame  in  favor  of  another.'^  The  same  possibility  of 
superposition  lies  open  as  we  go  forward  from  equation  (111)  to 

» See  Note  20. 


Reference  Frames  89 

consider  the  similar  transfer  for  accelerations,  though  the  com- 
plications soon  cut  down  any  advantage  of  a  verbal  expression 
for  it. 

Formal  routine  yields  for  the  time-derivative  of  the  general 
relation  in  equation  (110)  or  (111)  the  result 

^  =  to  +  (f  X  r')  +  2(y  X  v')  +  (y  X  (r  X  r'))  +  ♦',     (112) 

in  which  (r'),  (v'),  (v')  specify  the  position,  velocity  and  acceler- 
ation of  any  point  (Q)  by  means  of  (O',  X'Y'Z');  that  is,  to 
recapitulate, 

r'^iV  +  jy  +  kV;        v'^i'^  +  r^'  +  k'l^; 

.,_.,d^x;  d^/  dV 

^   ~  ^    dt^  "^  ^    dt2  "^      dt2   ' 


(113) 


(y)  is  the  angular  acceleration  belonging  at  the  epoch  to  the 
rotation-vector  (y),  and  (vo)  denotes  the  acceleration  of  (O') 
in  (0,  XYZ).  Interest  will  center  here  upon  the  terms  affected 
by  the  rotation,  into  which  the  elements  (r')  and  (v')  individual 
to  the  point  (Q)  enter;  and  for  the  latter,  the  connections  shown 
in  equation  (111)  must  be  duly  heeded.  It  will  cultivate  control 
of  details  in  the  method  to  carry  through  its  application  to  such 
combinations  as  (y  =  0),  (y  =  0)>  separately  or  conjointly,  in 
preparation  for  the  summary  that  follows.  And  then  to  work 
out  lists,  comparable  with  that  in  section  71,  for  the  general 
conditions  of  equations  (107,  111,  112),  showing  how  the  different 
quantities  are  affected  by  the  transfers  from  one  reference-frame 
to  another  that  have  been  brought  under  review.  It  is  always 
a  reciprocal  interdependence  that  is  in  question,  and  a  procedure 
for  transfer  in  either  direction. 

77.  To  round  out  this  stage  of  the  inquiry,  we  can  now  formu- 
late for  velocity  and  acceleration  the  suppositions  necessary  to 
their  invariance,  that  will  put  the  frames  for  which  these  are 


90  Fundamental  Equations  of  Dynamics 

satisfied  to  that  extent  on  an  equal  or  indifferent  footing.  We 
begin  with  acceleration,  whose  invariance  necessitates  con- 
formably to  equation  (112), 

^0  +  (y  X  r')  +  2(y  X  v')  +  (y  X  (r  X  r'))  =  0.       (114) 

But  (vo),  (y)  and  (y)  are  to  be  assumed  independently  of  each 
other;  and  further,  the  search  is  for  a  general  relation  covering 
all  points  (Q)  in  all  phases  of  their  motion,  which  puts  aside  as 
insufficient  every  particular  adjustment  or  singular  value  like 

r'  =  0;        v'  =  0; 

or  colinear  factors  in  some  individual  vector  products.  Hence 
the  proposed  invariance  of  acceleration  demands  all  three  con- 
ditions, 

to  =  0;        Y  =  0;        Y  =  0.  (115) 

These  permit  the  comparison-frame  to  have  unaccelerated  trans- 
lation with  (O'),  but  forbid  changes  in  orientation  (y)  as  indicated 
by  its  time-derivatives  of  the  first  and  second  order. 

The  invariance  of  velocity  imposes  different  limitations  deriv- 
able by  inspection  from  equation  (111)  as  being 

Vo  =  0;        Y  =  0.  (116) 

The  second  of  these  conditions,  therefore,  is  common  to  the 
invariance  of  velocity  and  of  acceleration.  But  as  regards  the 
translation  with  (O')  equation  (116)  excludes  any  velocity  (vo) 
though  allowing  an  acceleration  (to),  while  equation  (115)  inverts 
these  relations.  The  double  condition  for  invariance  of  velocity 
bars  at  the  epoch  motion  of  (O',  X'Y'Z')  in  (O,  XYZ),  but  gives 
freedom  as  to  subsequent  states.  The  triple  condition  for 
invariance  of  acceleration  maintains  the  exclusion  of  -changing 
orientation  and  sharpens  it  by  (y  =  0),  but  allows  any  constant 
value  of  the  vector  (vo). 

The  above  conclusions  coupled  with  the  discussion  that  led 


Reference  Frames  91 

to  equations  (98)  and  (100)  bring  out  how  (O',  X'Y'Z')  if  treated 

as  moving  in  the  standard  frame  must  always  sacrifice  in  some 

degree  the  invariant  properties  in  regard  to  velocity,  acceleration 

and  the  dynamical  quantities  dependent  upon  them;    though 

these  are,  nevertheless,  preserved  intact  by  a  succession  of  frames, 

each  in  coincidence  with  the  moving  frame  at  one  epoch.     The 

permanent  values  of  (ro)  and  (y)  for  the  stationary  frames  are 

marked  off,  one  by  one,  in  the  series  of  instantaneous  values  for 

those  elements  belonging  to  the  moving  frame.     In  this  sense 

and  to  this  extent,  the  presence  or  absence  of  an  invariance  that 

happens  to  be  in  question  can  be  made  to  turn  upon  the  point 

of  view,  which  because  it  affects  values  also  raises  issues  that  need 

to  be  decided  in  the  light  of  clear  statement  of  the  position  our 

thought  has  occupied.     Consequently  it  is  likely  to  repay  us, 

if  we  enforce  this  main  idea  by  approaching  it  in  reliance  upon 

the  frames  of  permanent  configuration,  the  mathematics  being 

modified  to  match. 

Invariant  Shift. 

78.  Whereas  the  radius-vectors  (r)  have  been  handled  in  the 
preceding  equations  as  functions  of  time  alone,  directly  in  (O, 
XYZ)  and  in  (O',  X'Y'Z')  through  the  relation 

r  =  ro  +  r',  (117) 

this  second  mode  of  making  a  beginning  will  disguise  the  same 
radius-vectors  (r)  into  functions  of  three  independent  variables 
(t.  To,  y).  And  this  will  evidently  lead  toward  fixing  attention 
upon  a  whole  group  of  comparison-frames  inclusively,  to  be 
constructed  by  assigning  continuous,  but  otherwise  arbitrary, 
values  to  (ro)  and  (y),  perhaps  in  connection  with  equation  (93), 
while  (t)  remaining  unchanged  gives  simultaneous  currency  to 
those  values. 

The  exact  differential  of  (r)  indicated  according  to  the  new 
terms  is 


92  Fundamental  Equations  of  Dynamics 

dr  =  -dt  +  — dro  +  T-d7.  (118) 

at  oTo  oy 

This  form  might  indeed  be  denominated  rather  sterile  of  meaning 
in  respect  to  (r)  itself,  for  it  is  apparent  enough  from  many  of 
the  expressions  that  we  have  been  laying  down  that  (r)  is  not 
intrinsically  dependent  on  either  (to)  or  (y).  Similarly  if  we 
use  equation  (117),  and  after  omitting  the  terms  that  are  neces- 
sarily zero,  on  our  assumption  about  independent  variables,  write 

appeal  to  equation  (94)  seems  to  tell  that  (r')  at  any  epoch  does 
not  change  with  (y).     But  after  admitting  that 

^  =  ^  +  ^  =  0;        T-  =  0;  120 

dro       OTo       OTo  dy 

equation  (119)  is  found,  notwithstanding,  really  helpful  for  the 
end  sought,  as  a  starting-point  for  collating  dijfferent  sets  of 
components  within  our  group  of  frames,  though  it  might  be 
superfluous  did  we  restrict  ourselves  to  resultants.  In  order  to 
develop  this  idea  more  fully  introduce  the  semi-cartesian  equiva- 
lent 

r'  =  i'x'  +  jy  +  k'z',  (121) 

whose  second  member  is  intended  for  a  comprehensive  notation 
applying  both  tensors  (x'y'z')  and  unit-vectors  (i'j'k')  generically 
to  the  whole  group.  They  are  then  variables  as  affected  by 
passage  from  one  frame  to  its  neighbors,  and  in  addition  the 
tensors  are  variable  with  time  in  the  same  frame. 

This  temporary  identity  of  the  variables  in  the  one  frame,  which 
may  pick  that  one  out  or  enable  us  to  recognize  it,  and  yet  be 
evanescent  for  the  group  of  frames  as  a  whole,  lies  close  to  the 
heart  of  the  thought  in  equation  (119),  as  contrasted  with  a 
completer  convection  of  identity  with  one  moving  frame,  whose 


Reference  Frames  93 

tensors  and  unit-vectors  are  consequently  functions  of  time  only. 
For  the  present  purpose,  on  the  other  hand,  and  in  its  adapted 
mathematics,  the  tensors  (x'y'z')  must  be  considered  functions 
of  (ro),  (y),  (t);  but  the  unit-vectors  (i'j'k')  and  (ro)  do  not  at 
this  stage  vary  by  mere  lapse  of  time ;  nor  the  former  by  reloca- 
tion of  the  origin  (O') — they  must  be  functions  of  (y)  alone. 
Under  the  suppositions  and  the  reasons  for  them  thus  made 
explicit,  we  execute  the  differentiation  of  equation  (121)  in 
combination  with  equation  (119)  and  obtain 


dr  =  T—  dr 

5ro 


,{dx'  ,         ax'  ,         dx'      \ 

(a^<*'«  +  57<^^  +  ir*) 

.  ,( Sz'   ,  dz'   ,  3z'       \ 


+  i 

+  j 


(122) 


+ 

79.  This  expansion  supplies  material  to  interpret  profitably, 
when  it  is  observed  that  the  imposed  condition  for  the  partial 
time-derivatives  with  the  set  of  variables  now  adopted  is  the 
same  in  effect  as  that  for  invariant  velocity  to  which  equation 
(97)  is  subject.  Consequently  the  three  terms  on  the  left  are 
properly  equated  to  the  velocity  of  any  (Q)  in  the  standard  frame, 
when  we  write 

The  double  use  of  this  equality  is  apparent,  either  in  obtaining 
projections  of  known  (v)  upon  the  (X'Y'Z')  of  the  configuration, 
or  in  determining  (v)  by  means  of  its  projections  upon  whatever 
particular  comparison-frame  is  designated  by  the  stationary 
values  at  which  (ro)  and  (y)  are  arrested  while  the  partial  change 
with  (t)  is  recorded. 


94  Fundamental  Equations  of  Dynamics 

Thus  no  essential  in  regard  to  consistent  expression  of  velocities 
would  be  sacrificed  if  we  depended  upon  any  such  comparison- 
frame  momentarily  to  replace  (O,  XYZ)  in  its  service  as  standard, 
and  did  likewise  for  new  stationary  values  of  (ro)  and  (y)  with 
velocities  at  other  epochs.  This  comment  will  infuse  its  due 
quota  of  meaning  into  the  equality 

f^v  =  ~  (124) 

and  parallel  expressions,  whenever  similar  opposed  total  deriva- 
tives and  partials  are  made  to  play  their  roles  as  the  basis  of  a 
regular  procedure,  in  which  a  resultant  vector  is  to  be  con- 
structed or  evaluated  by  means  of  components  parallel  to  axes 
that  differ  systematically,  or  in  which  the  projections  of  a  given 
vector  upon  such  axes  appear  naturally. 

It  is  readily  apprehended,  at  this  point,  how  such  plans  are 
effectively  equivalent  to  a  continuous  process  of  transfer  to  new 
standard  frames  that  is  kept  simple  by  its  preservation  of  invari- 
ance,  while  it  may  secure  a  permanence  of  form  or  other  ad- 
vantage in  addition.  The  indispensable  resolution  of  accelera- 
tion along  tangent  and  normal  of  the  epoch  in  treating  curved 
paths  is  one  case  in  point;  and  the  compact  forms  obtained  by 
introducing  principal  axes  will  suggest  strongly  some  similar 
scheme  in  continuation  of  sections  61  and  63  with  expectation  of 
profit  from  it.  It  seems  convenient  to  have  a  brief  name  for 
contrived  plans  of  this  character,  so  we  shall  refer  to  them  here- 
after as  shift  of  reference-frame,  implying  always  invariant  shift 
in  so  far  as  some  quantities  are  not  thereby  modified  from  the 
simultaneous  value  indicated  in  the  standard  frame.^ 

80.  The  three  terms  put  down  in  equation  (123)  are  then 
seen  to  reproduce  accurately  in  the  combinations  of  iequation 
(122)  the  actual  displacement  (dr)  for  the  time  (dt)  of  the 
moving  point  (Q)  in  the  standard  frame;    and  therefore,  the 

1  See  Note  21. 


Reference  Frames  95 

remaining  entries  in  the  coefficients  of  (i'j'k')  must  be  illusory 
if  taken  by  themselves,  as  regards  describing  what  is  thus 
happening  at  (Q).  In  fact,  as  their  form  involving  constancy 
of  (t)  indicates  clearly,  they  are  attendant  upon  comparisons  of 
corresponding  and  simultaneous  pairs  in  two  sets  of  projections 
determining  or  determined  by  the  same  (r'),  but  connected  with 
two  sets  of  axes  differing  in  orientation  by  (dy)  and  having 
origins  separated  by  (dro).  The  complete  coefficients  of  (i'j'k') 
being  evidently  the  exact  differentials  for  the  present  inde- 
pendent variables  of  the  tensors  (x'y'z'),  equation  (122)  can  be 
rewritten 

dr  =  dro  +  (dr  X  r')  +  (i'dx'  +  I'df  +  k'dz'),      (125) 

if  we  bring  in  the  consequences  of  the  rotation- vector  (dy)  in 
the  form 


(126) 


Accordingly  equation  (125)  in  its  second  member  is  so  arranged 
that  it  includes  within  its  last  group  deviations  from  the  true 
value  of  (dr)  through  apparent  or  spurious  changes  in  the 
tensors,  and  finally  offsets  these  by  the  corrective  first  and 
second  terms. 

That  exactly  the  compensating  adjustment  shown  must  exist, 
can  be  argued  summarily,  in  line  with  our  remark  upon  equations 
(119,  120),  from  the  independence  of  actual  -changes  in  (r)  of 
mere  subheadings  in  our  accounts  of  them,  but  some  few  details 
are  worth  inserting  for  emphasis.  The  first  of  equations  (120) 
is  self-evident,  for  (r')  must  lose  whatever  (lo)  gains,  while  (r) 
is  held  at  its  value  by  unchanging  (t).  Let  us  therefore  analyze 
only  the  second  of  those  equations  in  regard  to  the  dependence 
of  the  tensors  upon  (y).     We  must  have 


96  Fundamental  Equations  of  Dynamics 

x'  =  i'- (x  -  xo)  +  i'- (y  -  yo)  +  i'-  (z  -  zo).        (127) 

Then  because  neither  (xyz)  nor  (xoyoZo)  in  the  standard  frame 
are  dependent  upon  (y), 

—  d7  =  (^— d7  j-((x  -  Xo)  +  (y  -  yo)  +  (z  -  Zo)) 


Consequently 


(g'O 


(128) 


ax' 

—  d7  =  (dYxi')T'  =  -  (dYxr')-i';  (129) 

and  similarly 

^'d7  =  -  (dYxr')-j';  ^'d7  =  -  (dYxr')-k';  (130) 
which  together  prove  consistently  with  anticipation, 

81.  Let  us  next  return  to  equation  (110),  with  the  reminder 
that  it  occurs  in  a  general  procedure  of  substituting  a  new 
reference-frame  to  be  standard,  by  making  necessary  allowance 
for  the  relative  motion  of  the  two  frames.  Multiply  both 
members  by  (dt)  and  verify  that  its  form  then  becomes  identical 
with  equation  (125),  although  the  latter  was  deduced  under 
more  special  limitations  that  we  propose  to  distinguish  as  shift, 
and  that  keep  the  velocities  invariant.  In  other  words,  the 
sum  of  the  last  three  terms  in  this  equation  will  differ  by  the 
same  amount  from  an  actual  displacement  (dr)  in  the  standard 
frame,  whether  (dro)  and  (dY)  designate  differentially  changes 
of  configuration  observable  in  the  one  moving  comparison-frame, 
or  whether  the  same  elements  express  the  shift  in  passage  to  a 
consecutive  member  of  the  invariant  group  of  frames. 


Reference  Frames  97 

These  two  relations  distinct  in  their  conceived  source  are 
joined  into  a  formal  identity,  primarily  because  together  they 
embrace  a  series  of  coincidences,  as  displayed  in  sections  69  and 
77,  for  each  aspect  of  which  the  same  symbols  can  be  given 
coherent  meaning.  But  that  fact  though  patent  is  no  good 
ground  for  obliterating  either  one  of  the  serviceable  conceptions 
out  of  which  the  equation  that  we  are  now  discussing  has  arisen 
for  us.  We  should  rather  grasp  firmly  the  thought  that  two 
successions  are  here  instructively  coordinated:  one  ensuing  by 
movement  of  an  identified  frame  into  new  positions,  and  the 
other  by  timeless  shift  to  new  stationary  frames.  These  con- 
clusions refer  in  this  first  instance,  of  course,  only  to  the  velocities 
for  which  they  have  been  established ;  but  they  are  conveniently 
capable  of  extensions.  In  the  measure  that  these  are  unfolded, 
they  will  lend  finalh'^  to  the  otherwise  trivial  identity 

A  =  (A  -  B)  +  B  (132) 

that  equation  (125)  may  suggest,  a  value  for  working  needs 
through  practically  advantageous  selections  of  (B).  Note,  for 
example,  that  equation  (74)  is  scarcely  different  in  type. 

82.  As  the  last  remark  might  imply  somewhat  plainly,  the 
exploitation  of  the  dominating  idea  in  shift  will  look  to  govern 
its  course  and  its  extent  by  special  phases  of  adaptation  con- 
trived to  meet  combinations  that  do  occur.  Analysis  that  we 
shall  undertake  of  several  coordinate  systems  may  be  expected 
to  illustrate  and  repeat  that  lesson.  What  the  instances  quoted 
in  section  79  show  is  more  generally  true:  That  the  plans  for 
shift  require  various  adjustments  to  be  renewed  continuously, 
and  keep  modulated  pace  with  conditions  that  develop  velocity, 
acceleration  and  the  closely  related  dynamical  quantities.  Thus 
the  progress  of  the  shift  must  accommodate  itself  to  a  regulative 
time-series  of  other  values,  and  this  in  turn  imposes  upon  the 
shift  process  itself  a  necessary  rate  in  time.     That  situation  the 


98  Fundamental  Equations  of  Dynamics 

mathematics  handles  by  recognizing  (to)  and  (y)  to  be  functions 
of  time,  instead  of  treating  them  as  independent  variables  subject 
only  to  timeless  change;  so  linking  them  with  each  other  and 
with  the  salient  phenomena  that  are  to  be  followed  up  that  some 
hne  of  advantage  sought  is  most  nearly  secured. 

Nevertheless  since  the  previously  independent  increments  still 
form  a  background,  these  additional  functions  of  time  will  differ 
in  certain  respects  from  those  that  yield,  for  instance,  the  veloci- 
ties and  accelerations  of  the  moving  points  (Q) .  One  formulation 
of  the  critical  difference  declares  that  the  latter  class  of  time 
functions  is  dictated  altogether  by  an  objective  element;  they 
must  conform  to  the  phenomena  studied  and  express  them,  their 
own  nature  and  form  being  to  that  important  extent  not  under 
control.  Those  of  the  former  class  are  open  to  free  choice, 
although  we  may  grant,  indeed,  that  this  control  is  exercised 
normally  in  bringing  to  pass  some  mode  of  subordination  to  what 
is  occurring  in  other  sequences,  to  the  end  of  attaining  simpler 
models  in  equations,  or  the  like  removal  of  complications.  This 
employment  of  time  functions  in  dynamics  that  are  distinguish- 
able in  their  nature,  has  long  been  commented  upon  and  provided 
for,  though  the  discrimination  is  stated  variously  and  not  always 
in  clearest  terms.^ 

On  a  foundation  of  the  foregoing  explanation  or  some  equiva- 
lent, we  are  brought  to  accept  a  two-fold  dependence  upon  time 
in  equation  (122)  and  in  any  statements  that  disclose  to  examina- 
tion the  grounds  for  a  similar  distinction.  Thus  we  gain  the 
liberty  to  regard  the  partial  processes  as  simultaneous,  to  divide 
equation  (125)  by  (dt)  and  so  to  establish  an  exact  formal  identity 
with  equation  (110)  by  allowing  for  shift  rates  that  are  inde- 
pendently assignable.  Yet  the  alternative  readings  diverge  still 
in  the  direct  meanings  associated  with  (fo)  and  (y);  these  are 
alike,  however,  in  standing  equally  among  the  controllable  time 

J  See  Note  22. 


Reference  Frames  ,99 

rates,  because  the  one  definite  frame  to  which  transfer  shall  be 
executed  may  move  at  will,  save  as  outlook  toward  convenience 
guides  or  special  circumstances  demand.  Perhaps  it  is  not 
over-subtle  either  to  insist  upon  a  second  residual  difference: 
The  plan  of  equation  (110)  aims  primarily  to  connect  properly 
two  sets  of  values  for  velocity,  each  correct  and  complete  for  its 
own  conditions;  but  equation  (125),  on  the  contrary,  entertains 
only  one  set  of  values  as  correct,  that  are  made  to  reappear 
finally  from  being  obscured  under  a  transient  distortion. 

83.  We  should  not  have  elaborated  these  ideas  with  equal 
fullness  had  the  results  borne  solely  upon  the  narrower  issues 
gathered  about  the  radius-vector,  and  had  not  Hamilton's  hodo- 
graph  given  a  clew  toward  making  the  radius-vector  repre- 
sentative of  other  vectors,  and  the  velocity  of  its  extremity  a 
key  to  the  general  vector's  time  rate.  The  vector  algebra  having 
fallen  heir  to  these  methods  and  enlarged  them,  it  is  natural  to 
look  upon  the  previous  section  as  a  preface  and  proceed  to  trace 
again  its  characteristic  connections  when  any  vector  (V)  has 
replaced  (r),  and  its  time-derivatives  are  offered  in  parallel  with 
(v)  and  (^).  In  the  course  of  such  extension,  we  may  expect 
correspondences  and  fruitful  grafting  of  larger  ideas  upon  the 
parent  special  case,  all  along  the  line  of  development  whose 
details  are  now  fairly  before  us. 

But  when  we  come  to  examine  and  sort  the  material  that  pre- 
sents itself  under  such  headings,  we  find  the  two  chief  operations 
that  we  have  been  comparing  very  unequally  represented  in 
practice.  The  circumstances  of  unrestricted  change  from  one 
reference-frame  to  another  do  reappear  in  connection  with  all 
physical  vectors  and  other  types  of  quantity;  and  as  we  have 
seen  exemplified  repeatedly  already,  those  changes  when  they 
are  made  necessitate  a  deliberate  reconsideration  of  all  these 
quantitative  values.  Yet  besides,  the  occasions  that  compel 
such  revisions  are,  at  once,  comparatively  rare  and  apt  to  be 


100  Fundamental  Equations  of  Dynamics 

made  for  conditions  that  have  become  more  strongly  speciaHzed ; 
although  the  process  is  important  as  regards  flawless  execution, 
it  shows  few  features  that  give  it  the  weight  of  a  procedure  that 
holds  its  place  among  the  routine  methods  of  frequent  use. 

The  alternative  conception  that  we  call  shift,  however,  has 
been  introduced  and  given  preliminary  analysis  here  to  a  degree 
that  may  seem  not  quite  called  for,  because  in  the  first  place  it 
is  implicitly  or  explicitly  involved  when  a  number  of  the  standard 
coordinate  systems  in  dynamics  are  employed,  which  is  a  routine 
procedure;  and  because  secondly,  there  has  been  some  failure 
in  clear  apprehension  and  announcement  of  just  those  conse- 
quences of  the  restrictions  upon  the  process  of  shift  that  bring  it 
into  close  alliance  with  the  prevailing  purpose  of  coordinate 
systems.  For  these  are,  in  the  main,  adapted  to  the  one  central 
idea  of  expressing  equivalently  or  invariantly,  through  some 
convenient  dissection  into  parts,  a  resultant  or  total  quantity 
that  relations  in  a  standard  frame  have  first  actually  or  potentially 
settled  upon.  When  therefore  we  dismiss  in  a  few  sentences  the 
subject  of  changing  reference-frame  for  the  general  vector  (V), 
and  yet  expand  the  idea  of  shift  on  its  broader  lines,  the  explana- 
tion is  to  be  sought  in  the  reasons  that  have  just  been  given. 

84.  If  we  look  again  at  equation  (95)  with  a  view  to  generalizing 
upon  it,  we  must  describe  (ro)  as  the  difference  between  the 
values  in  the  two  frames  of  the  vector  that  is  under  considera- 
tion.    Similarly  if  we  write  the  equation 

V  =  Vo  +  V  (133) 

in  beginning  an  attempt  to  extend  the  validity  of  previous  con- 
clusions, it  is  clear  how  (Vo)  is  to  be  read.  It  is  also  apparent, 
or  verified  by  easiest  trial,  that  one  obstacle  to  indicating  here  a 
more  general  rule  for  change  of  reference-frame  enters  because 
the  value  of  (Vo)  depends  upon  the  quantity  represented  by  (V), 
as  instanced  by  the  conditions  for  invariance  in  section  77.     But 


Reference  Frames  101 

it  was  also  forced  upon  our  attention,  from  equation  (94)  onward, 
that  (r')  in  the  standard  frame  is  invariantly  given  by  all  frames 
whose  origin  is  at  (O')  in  its  position  for  the  epoch.  And  while 
this  too  draws  the  lines  closer  for  (V)  and  limits  narrowly  the 
usefulness  of  results  attached  to  derivatives  of  (r),  (ro)  and  (r'), 
in  doing  that  it  points  convincingly  toward  the  process  of  shift, 
if  we  are  to  generalize,  in  which  this  very  invariance  has  been 
made  a  prominent  characteristic.  When  we  look  at  the  matter 
from  another  side,  and  observe  how  near  an  assigned  behavior 
of  (i'j'k')  comes  to  furnishing  completely  the  compensating  or 
corrective  elements  in  an  equation  like  (125),  once  more  the 
conformity  of  a  coordinate  system  to  some  rule  of  displacement 
can  be  seen.  Thus  polar  coordinates  are  essentially  a  shifting 
orthogonal  set,  and  a  scrutiny  of  the  standard  expressions  for  the 
components  there  shows  that  they  meet  (r')  on  an  equal  footing 
of  reproducing  a  resultant  invariantly. 

85.  We  shall  begin  the  definite  inquiry  about  shift  in  its  larger 
relation  to  coordinate  systems  by  supposing  that  we  have  to  do  , 
with  any  free  vector  determined  in  the  standard  frame  as  (V), 
postponing  the  mention  of  locahzed  vectors.  Then  (V)  may  be 
associated  legitimately  with  the  origin  (O)  as  base-point,  and 
any  element  that  might  correspond  to  (ro)  will  be  suppressed. 
With  the  usual  unit-vectors,  here  taken  at  a  common  origin  for 
convenience,  we  must  have  at  the  epoch,  whatever  range  in 
orientation  may  be  permitted  for  (i'j'k'), 

V  =  iV(.)  4-  jV(,)  +  kV(.)  =  i'V(.')  +  j'V(y')  -f  k'V(.').  (134) 

This  relation,  to  repeat  with  emphasis  an  incidental  remark  of 
section  79,  may  face  in  either  of  two  directions,  according  as  the 
data  make  (V)  itself  or  its  three  constituents  directly  known. 
The  next  equation  derives  much  of  its  importance  from  the 
fact  that  the  algebra  so  seldom  furnishes  a  resultant  vector  im- 
mediately, unless  the  superficial  geometry  happens  to  fit. 


102  Fundamental  Equations  of  Dynamics 

Express  now  the  time-derivative  of  (V) ;  it  will  be  consistently 
specified  for  the  same  standard  frame  as  (V)  itself,  and  it  ap- 
pears as 

=  (i'V(.')  +  j'V(/)  +  k'V(.')) 

+  i'  ^  (V(.'))  +  j'  ^  (V(,'))  -I-  k'  ~  (V(.')).     (135) 

It  is  to  be  observed  about  tensors  like  (V(x'))  that  they  are 
differentiated  on  that  comprehensive  understanding  about  them, 
spoken  of  in  section  78,  which  is  favored  by  an  algebra  that 
attends  to  magnitudes  alone  and  can  neglect  orientation.  In 
the  first  group  of  the  third  member  in  this  equation,  it  is  the 
vector  algebra  with  its  equal  attention  to  directions  which  is 
repairing  that  deficiency  in  the  other  algebra.  In  order  to 
follow  up  and  express  this  idea,  we  adopt  the  notation  for  all 
such  cases, 

V(.)  -  i'  ^  (V(.'))  +  j'  ^  (V(,'))  +  k'  ^  (V(.')),    (136) 

intended  to  suggest  that  only  the  tensor  magnitudes  of  (i'j'k') 
have  been  dififerentiated.  Omitting  the  second  member  of  equa- 
tion (135),  and  in  reliance  upon  section  80  for  a  reduction  of 
the  first  group,  the  third  member  can  be  rewritten  in  the  more 
nearly  standard  form, 

V  =  (y  X  V)  +  V(„.).  (137) 

But  equation  (134)  would  not  be  modified  if  the  origin  for 
(i'j'k')  were  at  any  distance  (ro)  from  (0)  and  were  moving  in 
any  way.  Our  last  result  would  still  hold,  provided  the  same 
(y)  were  retained,  because  it  is  a  sheer  relation  for  projections 
upon  which  it  stands.  Further,  whenever  (y)  is  zero,  both  (V) 
and  (V)  are  represented  indifferently  by  their  respective  compon- 


Reference  Frames  103 

^ents  in  (XYZ)  or  in  (X' Y'Z') ;  and  this  harmonizes  with  the  invari- 
ance  found  by  using  the  permanent  configurations  of  the  coinci- 
dences and  the  idea  of  shift.  Otherwise  even  when  (i'j'k')  fall 
in  (ijk)  and  make  the  two  sets  of  components  for  (V)  the  same, 
the  total  time-derivatives  of  any  algebraic  expressions  for  the 
tensors  of  (i'j'k')  would  not  agree  with  the  projections  of  (V) 
on  (X'Y'Z').  But  note  that  the  proper  partial  derivatives  of 
those  tensors  would  give  correct  values  for  (V),  as  we  discovered 
from  equation  (123)  in  the  case  of  (v). 

There  is  one  condition  of  special  arrangement  that  cancels  the 
difference  between  (V)  and  (V(m))  though  (y)  is  not  zero;  namely, 
colinear  or  parallel  factors  in  the  corrective  vector  product.  And 
since  (y)  as  applying  to  (i'j'k')  rests  on  a  supposition  subject  to 
a  certain  control,  there  is  a  strong  hint  in  the  above  possibihty 
of  cancellation,  which  several  coordinate  systems  have  found 
their  own  ways  to  adopt.  We  can  give  a  first  illustration  from 
our  original  discussion  of  the  rotation-vector.  For  if  we  multiply 
equation  (137)  by  (dt)  and  identify  (V)  with  (dy)  the  two 
members  show  equality  to  the  second  order,  in  confirmation  of 
section  47. 

86.  Let  the  vector  (V)  be  represented  graphically  from  (O)  as  a 
base-point,  in  the  manner  of  the  velocity  vector  for  the  hodo- 
graph,  then  the  derivative  (V)  will  be  given  as  the  velocity  of  its 
extremity  in  (0,  XYZ);  and  on  comparing  equations  (111,  137), 
the  former  in  application  to  a  common  origin,  the  other  derivative 
(V(m))  is  seen  to  give  similarly  the  velocity  with  which  the 
extremity  of  (V)  moves  in  the  frame  (X'Y'Z').  Consequently 
we  find  forms  like  (V(m))  described  sometimes  as  derivatives 
relatively  to  the  moving  axes  (X'Y'Z'),  and,  to  be  sure,  they  are. 
But  we  must  not  neglect  the  other  fact  that  this  uncompleted 
derivative  is  applied  to  a  quantity  that  like  (V)  has  been  speci- 
fied for  the  standard  frame,  and  that  itself  does  not  stand  in  any 
one  particular  relation  to  the  frame  (X'Y'Z').     These  schemes. 


104  Fundamental  Equations  of  Dynamics 

if  thus  viewed,  are  composite;  or  they  straddle  between  the 
standard  frame  for  (V)  and  a  comparison-frame  for  (V(m)); 
but  they  are  less  disjointed  if  interpreted  as  shift.  The  above 
denial,  of  course,  runs  only  against  a  general  truth,  and  does 
not  exclude  special  conditions  under  which  the  same  term  covers 
both  a  shift  and  the  other  form  of  transfer.  It  is  plain  for 
example,  in  giving  velocity  by  means  of  polar  coordinates  in 
uniplanar  motion  as 

V  =  ri  ^  +  (<o  X  r),  (138) 

that  the  first  term  in  the  sum  can  be  read  either  as  (V(n,)),  or  as 
(v')  for  the  frame  consisting  of  (r)  and  a  perpendicular,  with  the 
second  term  equally  adapted  to  either  sense. 

It  contributes  much  to  the  serviceable  simplicity  of  equation 
(137)  that  it  observes  always  the  limits  of  a  one-step  transition 
from  a  vector  to  its  first  derivative,  while  a  radical  change  of 
reference-frame  must  rebuild  from  the  beginning  by  as  many 
steps  as  are  necessary.  Let  us  exemplify  how  contrasts  appear, 
by  taking  (v)  as  the  vector  of  equation  (137)  and  placing  the 
result  alongside  equation  (112),  from  which  (vo)  has  been  removed 
by  the  supposition  of  a  common  origin,  and  in  which,  for  closer 
parallelism,  we  have  substituted  for  (v')  in  terms  of  (v).  On 
one  hand  we  find 

v=  (y  XV) +♦(„.);  (139) 

and  on  the  other 

v  =  (y  X  r')  +  2(t  X  v)  -  (y  X  (y  X  r'))  +  v'.        (140) 

It  is  evident  how  the  latter  equation  has  accumulated  compli- 
cations in  its  two  steps  that  we  followed  earlier,  and  that  the 
last  terms  in  the  two  equations  are  not  reduced  to  equality  even 
by  making  (y)  constant. 

87.  With  this  exposition  accomplished,  of  the  consequences 
for  free  vectors  and  their  first  derivatives  of  their  inclusion  in 


Reference  Frames  105 

plans  of  shift,  we  can  proceed  to  add  for  localized  vectors  those 
supplementary  particulars  which  the  localizing  factor  makes 
necessary  in  relations  like 

(rxV)  =  (roxV)  +  (r'xV),  (141) 

when  account  is  taken  of  the  change  in  (ro)  due  to  shift  of  the 
comparison-frame  into  some  new  but  permanent  configuration. 
This  allowance  is  obviously  required  in  order  to  complete  the 
details  for  the  effective  momentary  replacement  of  (O,  XYZ) 
by  successive  members  in  the  group  (O',  X'Y'Z').  And  it  is 
most  easily  disentangled  from  other  elements,  by  using  that 
superposition  applying  to  similar  cases  which  was  indicated  as 
far  back  as  section  67. 

Using  the  temporary  notation 

M  =  (rxV);        M'^(r'xV);  (142) 

the  special  question  that  concerns  us  here  is  the  relation  between 
(M)  in  the  standard  frame  and  (M'),  the  latter  quantity  being 
expressed  under  the  guidance  of  ideas  that  it  will  be  well  to 
make  quite  explicit.  First,  the  vector  (V)  enters  both  products 
invariantly;  and  secondly,  its  total  time-derivative  appears 
without  distinction  in  both,  because  changes  in  (i'j'k')  being 
now  put  aside  in  order  to  consider  changes  in  (ro)  alone,  the 
corrective  term  of  equation  (137)  disappears.  But  thirdly,  with 
(y)  dropped  from  the  list  of  section  78  for  the  reason  named, 
(r')  becomes  a  function  of  the  two  variables  (ro,  t).  Then  its 
exact  differential  is  for  the  present  shift 

dr'  dr' 

and  if  this  is  timed  to  march  with  the  actual  changes  during  (dt) 
we  get 

d£'_ar^droar^  dro_.  ^_. 

dt  ~  aro  dt  "^  at  '        dt  ^  ^°'        at  ~  ^'        ^-^^^^ 


106  Fundamental  Equations  of  Dynamics 

the  last  equality  having  the  same  validity  as  in  equation  (124), 
Hence 


-(^XV) 


M'  =  (^  — xVj  +  (r'xV)  =  ((f-fo)xV)  +  (r'xV);     (145) 

M  =  (f  X  V)  +  (r  X  V)  =  M'  +  (ro  X  V)  +  (ro  x  V).     (146) 

Consequently,  though  (O')  coincides  with  (0),  if  there  is  dis- 
placement of  the  former  with  shift  rate  (fo)  the  values  of  (M) 
and  (MO  as  defined  will  still  differ  by  the  term  (to  x  V) . 
We  may  restate  the  last  equation  by  arriving  at  it  through 

M  -  M'  =  (ro  X  V);        M  -  M'  =  (h  x  V)  +  (ro  x  V),     (147) 

if  that  is  deemed  a  sufficient  analysis  of  the  conditions  for  the 
differentiation;  and  there  is  precedent  for  calling  (M')  the 
moment  of  (V)  for  a  moving  base-point.  It  is  only  iteration 
here,  however,  to  makie  the  comment  that  the  directer  thought 
holds  in  view  the  stationary  points  (0'),  for  which  the  coincident 
moving  point  serves  as  marker  at  beginning  and  end  of  the 
interval  (dt). 

Let  us  make  application  of  this  development  to  moment  of 
momentum  and  its  derivative,  as  being  the  localized  vectors 
among  our  fundamental  quantities.  We  are  still  confining 
attention  to  shift  of  origin  alone;  and  we  shall  not  go  beyond 
the  expressions  for  the  representative  particle  at  the  center  of 
mass.     Write  then 

H  =  (f  X  Q)  =  (ro  +  f)  X  Q  =  (ro  x  Q)  +  H(o');     (148) 
M'  =  ((V  -  ro)  X  Q)  +  (f  X  Q)  =  H(o'); 

and  reduce  by  omitting  the  product  of  colinear  factors.  But 
for  the  moment  about  (O')  of  the  force  measured  in  the  standard 
frame  we  have 

M(o')  =  f  X  Q  =  H(o')  +  (h  X  0),  (149) 


Reference  Frames  107 

which  thus  replaces  with  these  conditions  of  shift  the  relation  of 
equation  (VI). 

88.  For  estabhshing  the  theorem  of  equation  (137)  and  pre- 
senting its  bearings  and  a  few  of  its  consequences,  reliance  has 
been  placed  almost  exclusively  upon  the  vector  algebra;  yet 
those  ideas  were  manageable  to  the  other  algebra  also,  though 
it  cannot  fail  to  be  apparent  how  much  the  absence  there  of 
direct  indication  for  orientation  renders  the  operations  in 
matters  like  these  more  cumbrous,  and  the  expressed  results  less 
perspicuous.  If,  therefore,  it  seems  profitable  to  go  over  part  of 
that  ground  in  terms  of  the  older  method,  that  is  not  at  all 
with  wasted  effort  upon  verification,  nor  in  order  to  gain  reward 
in  fuller  insight,  except  as  seeing  the  cross  connections  is  likely 
to  prove  instructive.  But  coordinate  algebra  is  indispensable 
for  calculation;  transition  to  more  succinct  treatment,  where 
it  can  finally  displace  the  older  method,  is  still  in  progress,  which 
is  keeping  some  comparisons  temporarily  that  will  fall  away 
later;  and  moreover,  the  next  chapter  is  concerned  with  coordi- 
nate systems  as  its  chief  topic.  Consequently  in  preparation 
for  that  material  and  for  these  other  reasons,  it  seems  well  to 
put  in  a  link  of  connection ;  we  shall,  therefore,  proceed  to  parallel 
section  85  with  the  algebraic  equations  that  offer  the  same 
meaning  under  other  forms. 

It  is  unnecessary  to  carry  a  separation  of  origins  into  this 
development,  because  as  we  have  noticed  repeatedly  its  effects 
are  in  themselves  easy  to  record,  and  are  cared  for  completely 
by  uncomphcated  superposition.  Thinking  of  (X'Y'Z')  and 
(XYZ)  as  having  common  origin  (O),  (x'y'z')  and  (xyz)  are,  in 
the  first  instance,  the  coordinates  of  any  point  (Q).  But  we 
can  draw  advantage  in  two  ways  from  previous  experience; 
first,  (Q)  can  locate  a  representative  particle  of  finite  mass  as 
well  as  one  mass-element  of  a  body,  and  secondly,  (x'y'z')  and 
(xyz)  can  be  made  to  denote  the  projections  of  any  vector  (V) 


108 


Fundamental  Equations  of  Dynamics 


with  base-point  at  (0),  by  extension  of  their  relation  to  the 
particular  vector  (r)  that  is  now  identical  with  (r').  Unless  the 
contrary  is  said  explicitly,  (V)  is  to  be  regarded  as  determined 
in  the  standard  frame  (XYZ),  and  introduced  invariantly  into 
any  connections  with  (X'Y'Z').  This  vector  can  be  regarded 
as  localized  at  (0)  either  by  its  property  as  a  recognized  free 
vector  like  (Q)  and  (R),  or  by  a  convention  agreeing  with  its 
nature  in  cases  like  the  rotation-vector  (w)  and  its  companions 
(fa)),  (H),  and  (M)  when  pure  rotation  about  (O)  is  supposed. 
The  symbols  are  to  be  endowed  with  the  wider  valid  meanings  in 
the  equations  constructed  according  to  the  adjoining  table  that 
shows  the  direction  cosines  of  the  relative  configuration. 


89.  The  usual  transformation  equations  when  made  explicit 

for  (xyz)  are 

X  =  lix'  -I-  miy'  -1-  niz', " 

y  =  I2X'  +  may'  +  Uaz',  V  (150) 

z  =  I3X'  -1-  may'  +  ngz'.  . 

And  the  companion  forms  derivable  by  an  elementary  process 

are 

x'  =  lix  -1-  Uy  +  I3Z, 


=  miX  -H  moy  +  nisz,  - 
=  niX  +  n2y  -|-  Usz. 


(151) 


Together  these  are  known  to  depend  upon  or  to  express  the 
mutual  relations  of  projection  between  two  sets  of  components 
of  the  same  resultant  vector.     When  the  direction  cosines  are 


Reference  Frames  109 

invariable,  the  correspondence  with  constancy  of  (i'j'k')  is  evi- 
dent, and  the  same  mutual  relation  runs  on  into  all  the  deriva- 
tives, giving  invariance  whose  obvious  details  need  not  detain  us. 
A  change  of  configuration,  however,  makes  in  general  all  the 
direction  cosines  vary,  and  there  the  same  alternatives  recur 
that  were  brought  out  in  sections  78  and  82.  One  of  these  will 
make  (x',  y',  z')  each  a  function  of  three  independent  variables 
that  are  time  and  two  direction  cosines,  the  third  of  the  latter 
being  removed  by  a  standard  connection  like 

li^  +  h'  +  13=^  =  1.  (152) 

The  second  point  of  view  will  set  time  in  its  place  as  the  one 
independent  variable  of  which  all  other  quantities  are  functions; 
but  here  it  will  be  just  as  desirable  as  before  to  put  into  properly 
conspicuous  relief  the  modified  relation  of  time  to  variables 
like  (x,  y,  z)  and  to  others  like  (li,  la,  I3). 
90.  Equations  of  the  same  type  as 

:^-^^dt+^^d^  +  ^^dt  ^^^^^ 

can  be  read  in  the  light  of  equation  (123) ;  and  what  remain  to  ex- 
amine are  the  complete  time-derivatives  of  the  quantities  (x'y'z'), 
principally  in  order  to  detect  the  rotation-vector  (y)  of  (X'Y'Z') 
by  penetrating  its  disguise  of  direction  angles  and  their  deriva- 
tives. Adopting  the  fluxion  notation,  for  ease  in  writing  total 
time-derivatives,  we  have  first 

x'  =  (lii  -\-  Isy  -\-  I3Z)  +  (iix  +  Uy  +  isz).         (154a) 

Note  in  passing,  as  consequences  of  equations  (151,  154)  that 
may  prove  suggestive  later,^ 

ax'     dk\      J      d(dx'\_di' 

which  are  typical  of  similar  relations   running  all  through  the 
1  See  Note  23. 


110  Fundamental  Equations  of  Dynamics 

sets  of  equations,  when  we  add  to  the  value  of  (x')  its  com- 
panions 

y'  =  (miX  +  may  +  mgz)  +  (liiix  +  may  +  msz),] 
i'  =  (nix  +  nay  +  Usz)  +  (riix  +  nay  +  hsz).        J 

Concentrating  attention  upon  the  last  groups  in  these  equations, 
because  the  effects  of  changing  configuration  appear  exclusively 
in  them,  and  introducing  the  necessary  direction  angles  in  order 
to  prepare  for  the  connection  with  (y),  expand  into  the  forms 

—  [xdi  sin  ai  +  yda  sin  aa  +  zda  sin  as];! 

—  [x/3i  sin  /3i  +  yi32  sin  ^2  +  z^s  sin  183];  [  (156) 

—  [xei  sin  ei  +  yea  sin  a  +  zks  sin  €3].    J 

But  the  normal  to  the  plane  (X',  X)  must  be  the  axis  for  (di); 
and  with  the  direction  cosines  of  those  intersecting  lines  given  as 

1,  0,  0,  (X);        h,  I2,  I3,  (XO;  (157) 

the  direction  cosines  (X,  /i,  j')  of  the  normal  to  their  plane  worked 
out  by  the  standard  method  gives 

cos  as  cos  oi2 

X  =  0;        fi=-~. ;        v=-. .  (158) 

sm  ax  sm  ai 

But  as  explained  in  section  46  the  rate  at  which  (X')  is  turning 
about  that  normal  must  be  the  projection  of  (y)  upon  that  line, 
or  equivalently, 

dx  =  X7(x)  +  M7(y)  +  »'7(z),  (159) 

from  which  follows 

—  di  sin  ai  =  7(y)  cos  as  —  y^.^,)  cos  ag.  (160) 

Proceeding  similarly  with  the  eight  other  terms  which  complete 
the  group  of  that  type  in  equations  (154),  it  is  seen  after  simple 
reduction  that  they  make  up  in  the  first,  second  and  third  equa- 
tion respectively 

-   (7(y')Z'  -  7(z')yO;  -   (7(z')X'   -  7(x')Z' 

-   (7(x')y     -  7(y')X'). 


Reference  Frames  111 

Since  the  first  members  of  those  equations  correspond  to  the 
total  derivatives  of  the  tensors  obtainable  from  equation  (125), 
we  find  after  orientation  and  forming  the  vector  sum  that  equa- 
tions (154)  jdeld  consistently  with  equation  (137) 

V(m)  =  V-(yxV),  (162) 

on  our  understanding  about  the  broader  meaning  of  (x'y'z') 
and  (xyz). 

It  is  left  as  an  exercise,  modeled  on  the  above  plan  but  con- 
tinued into  the  formation  of  second  derivatives,  to  reach  by  the 
algebraic  routine  the  coordinate  equations  which  together  repre- 
sent the  result  recorded  in  equation  (112),  if  we  suppress  there 
all  terms  depending  on  a  separation  of  origins.  Where  the 
quantity  (y)  occurs  in  executing  this,  it  is  of  interest  to  reahze 
what  has  been  alluded  to  elsewhere;  that  (y)  and  (y)  may  be 
connected  with  either  (XYZ)  or  (X'Y'Z'),  since  the  difference 
term  in  equation  (162)  is  zero  when  (y)  is  (V). 


CHAPTER  IV 

The  Main  Coordinate  Systems 

91.  The  standard  frame  itself  has  an  additional  office  of 
providing  a  coordinate  system  that  is  basic  in  certain  ways,  and 
that  is  in  fact  tacitly  utihzed  for  the  semi-cartesian  expansions 
in  terms  of  (ijk) ,  both  in  immediate  relation  to  vector  quantities, 
and  for  the  expression  of  constituents  in  work,  kinetic  energy 
and  power,  where  vector  factors  occur  in  scalar  products.  To  do 
these  things  has  become  so  much  habitual  or  even  instinctive 
that  we  learn  with  some  surprise  how  Maclaurin  is  given  credit 
for  invention  here,  as  Euler  is  for  inventing  the  concept  of  fluid 
pressure,  which  at  this  date  might  also  seem  part  of  external 
nature. 

The  standard  frame,  too,  has  one  lead  in  advantage  over  other 
resolutions  through  the  unqualified  permanence  of  its  origin 
and  of  its  unit-vectors,  which  enables  us  to  submit  its  tensors 
unhesitatingly  to  algebraic  operations,  and  pass  over  to  vector 
algebra  by  merely  supplying  the  ellipsis  of  the  unaffected  ori- 
enting factors.  The  disturbing  influences  in  other  combina- 
tions, where  (ro)  and  (i'j'k')  make  more  caution  advisable,  have 
been  forcing  themselves  upon  us  repeatedly.  But  as  we  have 
seen  illustrated  for  mean  values,  and  as  is  not  unusual,  the 
presence  of  such  desirable  elements  as  we  find  in  the  standard 
frame  may  be  also  a  drawback.  Within  the  complete  projection 
on  a  standard  axis,  distinctions  of  source  in  changes  of  magnitude 
or  of  direction  may  be  lost,  that  are  vital  in  the  vectors  that  play 
a  part.  The  net  force  parallel  to  (X)  and  its  work,  if  written  for 
a  particle 

X  =  m^ ;        W  =  /Xdx;  (163) 

112 


The  Main  Coordinate  Systems  113 

hide,  in  the  first,  the  fact  that  normal  force  (N)  and  tangential 
force  (T)  are  coalescing  in  the  one  sum,  and  in  the  second,  that 
part  of  this  work  is  illusory  in  so  far  as  the  projection  of  (N) 
enters  the  sum  (X),  and  does  work  in  the  algebra  though  not  in 
the  mechanics.  At  one  other  point  we  have  been  enabled  to 
compare  the  principal  axes  of  inertia  with  (XYZ)  and  ascertain 
that  all  advantage  does  not  lie  with  the  latter,  for  expressing 
compactly  either  the  scalar  energy  or  the  vector  force-moment. 
And  these  considerations,  in  sum,  may  justify  us  in  leaving  the 
resolution  into  constituents  according  to  the  standard  axes  to 
one  side,  except  where  we  touch  upon  it  for  some  special  con- 
nection. Then  we  are  free  to  devote  detailed  attention  to  other 
coordinate  systems  that  are  chiefly  current,  and  make  due 
analysis  of  their  intention  and  of  the  scope  of  their  success. 

It  seems  quite  enough  therefore  if  we  collect  here  the  indicated 
partitions  for  (XYZ)  that  are  reasonably  self-evident  rewritings 
of  the  totals  to  which  the  preceding  text  has  given  most  weight: 

Q  =  i2/„,xdm  +  j2/„,ydm  +  kZ/„,zdm;  (164) 
H  =  iS/m(yz  —  zy)dm  -f-  j2/m(zx  —  xz)dm 

+  k2/„.(xy  -  yi)dm;  (165) 

E  =  iZ/^nxMm  +  i2/™y2dm  +  IS^zMm;  (166) 

R  =  i-Efmxdm  +  j2/„.ydm  -f  kS^zdm;  (167) 
M  =  iXfmiy'z  —  zy)dm  +  jS/m(zx  —  xz)dm 

+  kS/n,(xy  -  yx)dm;  (168) 

P  =  ^fmidX  +  S/n.ydY  -f-  2/„.zdZ.  (169) 

It  will  be  found  profitable  to  compare  equations  (165)  and  (86); 
also  equations  (166)  and  (81,  88),  including  the  comment  preced- 
ing the  latter.  Since  the  first  three  equations  in  the  above  group 
are  mere  expansions  of  the  forms  in  section  15,  they  have  the 
same  scope  as  those.     Similarly  the  validity  of  the  last  three  is 


1 14  Fundamental  Equations  of  Dynamics 

coextensive  with  that  for  equations  (16, 17, 18)  of  which  they  are 
the  expansions. 

Euler's  Configuration  Angles. 

92.  Because  it  deals  directly  and  exclusively  with  the  recurrent 
element  that  is  found  at  the  root  of  so  many  particular  results, 
we  shall  take  up  next  those  orientation  angles  for  specifying 
configuration  which  were  devised  by  Euler  and  by  custom  bear 
his  name.  They  have  not  yet  been  displaced  from  a  conceded 
position  of  value  in  use  for  their  purpose.  There  is  an  added 
reason  for  giving  these  angles  proper  discussion  in  that  the 
expression  of  them  as  vectors  has  scarcely  been  attempted;  we 
find  their  connections  with  other  specifying  elements  almost 
exclusively  in  the  form  of  purely  algebraic  equations.  It  is  a 
curious  fact  that  angle  in  prevailing  practice  has  not  arrived  at 
legal  recognition  as  a  vector,  though  the  vector  quality  of  its 
first  and  second  time-derivatives,  angular  velocity  and  angular 
acceleration,  was  announced  and  employed  a  number  of  years 
ago.  So  we  need  to  do  something  consciously  toward  incor- 
porating angle-vectors  into  our  scheme  of  treatment  on  a  parity 
with  other  vector  quantities,  in  order  that  real  symmetries  of 
relation  may  not  be  seen  distorted. 

Supposing  that  one  end  of  a  line  (r)  is  fixed  and  that  it  moves 
into  a  new  position,  its  second  configuration  in  relation  to  its 
first  can  be  given  by  a  vector-angle  normal  to  the  plane  of  the 
two  positions.  This  vector  is  axial,  and  related  to  an  area  with 
duly  assigned  circulation;  and  the  area  is  in  the  plane  located 
by  the  extreme  positions  of  (r),  its  magnitude  being  twice  that 
of  the  sector  of  the  unit  circle  limited  by  those  positions.  But 
such  a  direct  representation  of  this  total  would  be  no  more 
convenient  for  use  in  all  cases  than  other  resultants  are,  so  its 
projections  according  to  Euler's  plan  are  substituted,  which 
amounts  to  giving  the  latitude  and  the  longitude  on  unit  sphere 


The  Main  Coordinate  Systems  115 

centered  at  the  fixed  point  or  origin  (O),  in  which  (r)  cuts  that 
surface.  Assuming  next  that  (r)  is  a  definite  fine  of  a  rigid  soHd 
that  is  Kmited  to  pure  rotation  about  (O),  a  third  angle  added 
will  enable  us  to  complete  the  description  of  a  new  configuration 
for  the  solid,  and  this  last  angle  will  denote  a  rotational  dis- 
placement about  (r).  We  shall  follow  usage  in  assigning  the 
symbols  {■&)  to  the  latitude  angle,  and  (t|f)  to  the  longitude  angle, 
while  (^)  is  added  for  the  rotation  about  (r) ;  it  remains  only  to 
agree  upon  zero  values  of  the  three  angular  coordinates.  It 
suits  our  purpose  in  its  general  course  better,  to  think  in  terms 
of  a  displaced  rigid  cross  (X'Y'Z'),  which  may  here  be  made 
equivalent  to  the  rigid  solid  named  above,  and  then  coincidence 
of  (X'Y'Z')  with  (XYZ)  yields  the  natural  zero.  We  identify 
(Z)  with  the  earth's  polar  axis  in  its  relation  to  latitude  and 
longitude. 

93.  Beginning  with  resultant  angular  displacement  (y)  at  zero, 
and  (X'Y'Z')  coincident  with  (XYZ),  let  the  plane  (Y'Z') 
separate  from  (YZ)  by  angular  displacement  (t|f)  about  (Z),  in 
which  that  vector  angle  must  then  fall.  Next  let  angular  dis- 
placement (■&)  occur  about  the  displaced  position  of  (X'),  in  whose 
line  therefore  it  must  lie  as  a  vector  angle ;  and  finally  let  (X',  Y') 
turn  with  angular  displacement  (^)  about  the  final  position 
of  (Z'),  with  whose  line  this  third  vector  angle  must  then 
coincide.  To  make  the  conditions  standard,  (t{r,  d,  ^)  are 
all  to  be  taken  positive  by  the  rule  of  the  right-handed  cycle. 
The  order  of  the  three  displacements  has  been  chosen  so  that 
each  is  made  about  one  of  the  three  axes  (X'Y'Z')  as  found  at 
the  beginning  of  that  stage.  It  is  verified  without  difiiculty 
that  the  summed  projections  on  (XYZ)  are 

Y(x)  =  i(t?  cos  rp  -\-  <p  sin  d  sin  ^);  1 

T(y)  =  J(^  sin  ^  —  (p  sin  i}  cos  ^);  >■  (170) 

Y(,)  =  k(\(/  +  <p  cos  ^).  J 


116  Fundamental  Equations  of  Dynamics 

And  if  we  resolve  on  the  final  orientations  of  (X'Y'Z'),  those 
projections  are 


Y(x')  =  i'(i?  cos  <p  -\-  ip  sin  d^  sin  cp) ; 
Y(y')  =  j'(— t?  sin  (p  -h  ^  sin  i?  cos  cp); 
Y(,')  =  k'(<p  +  \P  cos  t?). 


(171) 


These  two  sets  of  projections  are  orthogonal;  but  if  we  state  the 
supposed  displacements  directly,  and  let  (t|fi,  ^i,  ^i)  represent 
unit-vectors  agreeing  with  those  suppositions,  the  set  is  obUque 
to  the  extent  that  the  angle  (t|fi,  ^i)  is  (■&)  and  not  in  general  a 
right  angle.     We  add  accordingly, 

Y  =  i^i('A)  +  ^iW  +  ^i(^),  (17^) 

and  have  secured  three  equivalent  forms  of  expression  for  the 
resultant  angle-vector  (y).  Observe  also  the  differences  among 
the  three  in  regard  to  the  unit-vectors;  (ijk)  are  permanently 
oriented,  (i'j'k')  are  capable  of  displacement  by  rotation,  for  they 
remain  orthogonal,  but  (tlfi,  ^i,  ^i)  must  be  considered  indi- 
vidually. It  is  seen,  if  we  hold  definitely  to  the  terms  of  the 
description,  that  (i|fi)  is  of  permanent  orientation  in  (Z),  that 
(■&i)  depends  for  orientation  upon  (i|r),  being  always  normal  to 
the  displaced  position  of  the  (Y'Z')  plane,  and  that  (^i)  depends 
similarly  upon  both  (i|r)  and  (■&),  because  the  (^)  displacement 
begins  where  the  second  stage  leaves  off.  All  three  quantities 
(ijf,  '&,  ^)  are  rotation-vectors  applying  to  the  axis-set  (X'Y'Z') 
as  representative  of  a  rigid  body,  and  standing  to  the  changes  of 
direction  of  individual  lines  in  the  relation  established  by  sec- 
tion 46.  This  needs  to  be  borne  in  mind  if  any  question  should 
be  opened  about  changing  the  sequence  of  the  three  steps,  so 
that  (p)  and  (^)  though  equal  to  their  first  magnitudes  are  con- 
nected as  vectors  with  different  axes. 

The  above  forms  of  statement  are  mathematically  on  the  same 
footing  as  a  means  of  determining  (y),  but  there  can  be  no  real 


The  Main  Coordinate  Systems  117 

doubt  where  the  preference  would  fall  on  the  score  of  ease  in 
application  or  execution,  when  the  three  plans  are  compared. 
The  second  is  especially  intricate  because  its  projections  are 
associated  with  that  very  terminal  configuration  of  (X'Y'Z') 
which  it  may  be  the  object  to  locate,  but  which  must  somehow 
become  known  before  the  scheme  can  assume  full  definiteness. 
It  should  be  inserted  however  for  the  sake  of  its  subsequent 
uses. 

94.  The  employment  of  the  standard  angles  (ijr,  ^,  ^)  is  not 
confined  to  expressing  configurations,  and  is  therefore  not 
exhausted  in  equations  (170,  171,  172).  Indeed  the  primary 
service  of  Euler's  so-called  geometrical  equations  has  begun  at 
their  developed  connections  with  the  rotation-vector  or  angular 
velocity,  and  found  a  natural  continuation  in  deahng  with 
angular  acceleration  written  (y)  or  (w).  As  we  now  undertake 
to  make  those  connections  clear,  combinations  will  occur  at 
first  or  in  later  application,  that  make  it  advisable  to  retain  (y) 
and  (y)  for  use  with  comparison-frames  hke  (X'Y'Z'),  and  let 
the  meaning  of  the  parallel  quantities  (w)  and  (w)  refer  ex- 
clusively, as  in  sections  45,  55,  62  and  63,  to  a  rigid  body's  rota- 
tion, either  about  its  center  of  mass  or  about  some  fixed  point. 
To  maintain  this  consistent  distinction  will  avoid  confusion  where 
both  pairs  of  elements  are  presented  in  the  same  inquiry. 

The  expressions  for  (y)  that  we  have  just  obtained  are  con- 
trived to  show  its  value  at  the  advancing  front  of  a  progressive 
angular  displacement  to  which  (i|r,  #,  ^)  can  be  considered  to 
belong.  Consequently  it  is  adapted  to  differentiation,  with  a 
view  to  exhibit  either  a  systematic  succession  of  partial  differ- 
entials or  simultaneous  time  rates  in  a  total  derivative;  and 
previous  discussions  have  laid  a  foundation  for  interpretations 
leading  in  both  directions.  In  the  first  instance  we  are  most 
nearly  concerned  with  the  derivation  of  (y)  from  the  three  several 
equations  (170,  171,  172)  and  the  collation  of  results  with  sec- 


118  Fundamental  Equations  of  Dynamics 

tion  85  as  bearing  upon  the  current  algebraic  forms.  And  because 
this  has  some  Httle  flavor  of  revising  the  latter,  the  fuller  infusion 
of  vector  peculiarities  into  these  matters  having  not  yet  worn  off 
its  novelty,  there  seems  to  exist  a  stronger  reason  for  detail, 
than  the  mere  arrival  at  conclusions  for  handy  use  might 
demand.^ 

95.  As  in  similar  comparisons  elsewhere,  the  (ijk)  projections 
furnish  reliably  through  pure  and  total  tensor  differentiation  an 
unquestioned  standard  to  which  alternatives  must  conform  if 
correctly  formulated.  So  the  first  straightforward  step  is  to 
employ  equation  (170)  in  this  test;  and  we  prepare  the  way 
with  the  expansion 


r/di?  d^   .         .       \ 

"^  ""  M  I  dt  ^^^  "^  "^  dt"  ^^°  '^  ^^^  ^  J 


(- 


+  (    —  r}  sm  \p  -rr  -\-  (p  COS  i?  sm  i/'  -77 


d^l^ 
+  (p  Sm  i9  COS  xf/  TT 


)] 


r/dt?  d<p  \ 

+  jN  ^sin^  -  ^sm^cosxl^j  (173) 

d^  dt? 

+  [  ^  cos  ^  37  —   ^  cos  I?  COS  ^  -77 


(• 

+  ^  sm  ??  sm  1/'  -7-  I 

But  we  have  been  remarking  from  section  79  onward  that  the 
partial  time-derivatives  in  equations  like  (171,  172),  when  the 
unit-vectors  are  made  variables,  must  reproduce  the  standard 
frame  values  obtained  through  (ijk).     Let  us  accordingly  write 

1  See  Note  24. 


(174) 


The  Main  Coordinats  Systems  119 

out  those  two  sets  of  partials  and  proceed  toward  comparing 
them  with  equation  (173).  Observing  that  the  conditions  of 
the  differentiation  exclude  trigonometric  functions  of  the  angles 
from  varying,  though  they  permit  the  angles  as  magnitudes  to 
change,  we  find 

^(i',j',k')=i  (^-cos^  +  -sin^sin<,j 

.,  /      5t?    .  3^    .  \ 

+  J  I  -  —  sm  (p  +  —  sm  I?  cos  ^  j 

^^..,....)  =  ..(^)  +  «,(f)  +  *.(^^).     (175) 

The  value  directly  apparent  in  the  last  equation  can  be  noticed 
by  inspection  to  agree  with  that  of  the  equation  preceding,  if  we 
assemble  mentally  from  the  latter  the  items  falling  respectiveh' 
along  (ill,  ^1,  ^i).  And  this  coincidence  is  next  to  be  recognized 
similarly  in  the  first  groups  marked  off  under  (i,  j,  k)  in  equation 
(173),  with  the  single  variation  that  the  latter  appear  as  total 
derivatives  of  the  angle  magnitudes.  The  patent  conclusion  is 
that  proper  allowance  for  the  difference  between  these  total  and 
these  partial  time-derivatives  must  exactly  offset  the  remaining 
groups  in  equation  (173);  and  that  outcome  might  be  accepted 
on  the  fair  ground  that  it  harmonizes  with  equations  (126,  131), 
without  going  further.  Yet  the  completed  analysis  of  how 
that  compensation  is  in  fact  brought  about  here,  has  an  im- 
mediate bearing  and  interest  that  justify  setting  down  its  several 
steps. 

96.  The  last  groups  of  terms  in  equation  (173)  can  be  brought 
together  and  rearranged  so  that  they  are  identified  as  the  vector 
products  to  which  they  are  equated  below: 
9 


120  Fundamental  Equations  of  Dynamics 

t?  -T-  (—  i  sin  ^  +  j  cos  rp) 

+  9?  -7-  (i  cos  7}  sm  ^  —  j  cos  t?  cos  (^  —  k  sin  z>) 

d^  ,.    .  .    .  .       N 

+  v?  TT  (1  sin  ??  cos  \l/  +  }  sm  1}  sm  ^) 

d^  dt?  dr^ 

=  ^  t?(tl:i  X  *i)  +  dt  ^^*'  ^  *'^  "^  dt  *'^''^'  ^  *'^' 

The  verification  as  regards  magnitudes,  directions  and  order  of 
factors  in  the  vector  products  is  ordinary  routine  devoid  of  arti- 
fice, due  regard  being  paid  to  the  specifications  of  direction  in 
the  sections  immediately  preceding.  The  character*  of  the 
second  member  is  plain:  it  consists  of  allowances  for  changing 
directions  of  the  two  unit-vectors  (^i)  and  (^1),  the  former  being 
affected  by  the  turning  about  (i|ri),  and  the  latter  by  the  two 
turnings  about  (1^1)  and  (*i).  It  is  instructive  to  notice  that 
these  individual  consequences  of  the  changes  in  the  unit-vectors 
preserve  their  type  and  enter  singly  in  parallel  with  the  develop- 
ments of  sections  47  and  80,  although  there  is  here  no  common 
factor,  the  rotation-vector,  related  equally  to  all  three  unit- 
vectors  (tti,  di,  ^1).  This  line  of  attack  has  been  adopted  partly 
in  order  to  extend  in  that  direction  our  earlier  proof. 

In  preparing  to  demonstrate  that  the  differences  between 
(ai?/at)  and  (dt?/dt),  (dcp/dt)  and  (d^/dt),  exactly  nullify  the 
second  member  of  equation  (176),  it  is  most  direct  to  start  from 
explicit  values  of  {\l/,  t?,  <p).  By  a  process  of  elementary  elimina- 
tion applied  to  equations  (170)  it  foUows  that 


cos  ^ 
sm  ir 

t?  =  7w  COS  ^p  +  7(y)  sin  \p; 

1 


<P  = 


sin  t? 


(7(x)  sin  xp  -  7(y)  COS  rp). 


(177) 


The  Main  Coordinate  Systems 


121 


It  is  to  be  remarked  as  regards  these  equations  that  in  order  to 
arrive  at  their  partial  time-derivatives,  we  must  include  as 
variables  only  (7(r)),  (7(y)),  (7(»))>  and  for  the  total  derivatives 
we  must  include  also  all  the  other  factors  as  functions  of  time. 
It  is  therefore  possible  to  write  these  indications  of  the  differences : 

d^ 

dt 

di?  _  M  _  Md^ 

dt  ~  at  ~  a^  dt  ' 

d(p      dip      d<p  d\l/      d<pd^ 

dt~dt^d4'dt~^d^dt 
Evaluating  the  second  members  from  equations  (177)  and  finally 
adding  the  orienting  unit-vectors  we  derive  these  expressions: 


a^A  _  ^dr/;      d4^d^ 

dt  ~  a^  dt  "*"  at?  dt 


(178) 


^^Pd^P 

"^'aiAdt 


cos  t?  .         d^ 

-  ^igiu^  WW  cos  '^  +  T'cy^  s^°  '^)  dt 


(cos  T?    d\l/\ 
"sln^^dt  j' 


ai/'  di?  1      /  .     ,  X  di^ 

^^^  dt"  =  ^^sln^  ^^^^^  '^°  "^  -  '''''  '^'  ^^  d? 

1^ 
sin  7? 
d«A 


/     1         dt?\ 
^^^l^In^^dtj' 


^1^  ^  =  -^U-  7(x)  sm  i/'  ^-  7(y)  cos  ^)  ^^ 


=  i».(-^sin,?^); 


a^  d^ 
ai/'dt" 


1^ 

sin  t? 


?i  nir^  (7(x)  cos  r/'  -f-  7(y)  sin  ^) 


di/^ 
dt 


-^^(sint^'^dt)' 


cos  t?  ,  .  X  \  d?? 

,i^3^(T(.)sm^-7(.)Cos,A)j^ 


_       /      cost?     dt?\ 
-^^V~sln^''dt;-J 


(179) 


122 


Fundamental  Equations  of  Dynamics 


After  forming  them  into  three  groups  as  shown  below,  they  can 
be  recognized  as  constituting  the  vector  products  to  which  they 
are  severally  equated ; 


'H 


diA 
dt 


cost? 


'  sin  t?  y 


d^ 


dt 


rHiixi^i); 


<P 


d^ 
dt 


/         1  cost?\  di?     .  . 

l^^si^-^^s"In^J=-dt^^*^^^^^' 

diA  .  diA    , 

-  ^Kp  ^-  sm  I?  =  -  ^-  (p{^i  X  ^i). 


(180) 


The  first  quantity  of  these  three  is  known  by  the  first  parenthesis 
to  be  perpendicular  to  (tl;i)  in  the  plane  of  (tjfi,  ^i) ;  so  the  second 
quantity  is  perpendicular  to  (^i)  in  the  same  plane;  and  (fl^i) 
is  by  supposition  normal  to  that  plane.  The  directions  match 
the  order  of  factors  and  the  signs. 

97.  When  the  established  conclusions  of  equations  (176,  180) 
are  united  with  what  was  found  to  be  true  on  casting  up  into  a 
vector  sum  the  three  first  groups  in  the  coefficients  of  (i,  j,  k), 
equation  (173),  the  registration  of  all  these  connections  yields 
the  continued  equality 


.  { d^  dip    .  .        \ 

=  1  I  —  cos  ^  +  —  sm  t?  sm  i/-  I 

.{d^.  dip    .  \  .  .   f  dxf^       d<p  A 

+  ^ U  ''"'  ^"  eJ '^^  ^  '^'  V-^^[dt  +  at  ^^'  V 

=  i'(^-cos^  +  -sm.?sm^j 
+  jM  -  — -  sm  ^  +  —  sm  ??  cos  ^  I 


(181) 


The  Main  Coordinate  Systems  123 

dip  d&  dip      (      d\p  d??  d^\ 

drl/  d\t/  d^ 

The  last  member  is  a  specially  plain  demand  of  the  vector  algebra, 
in  order  to  reconcile  the  value  of  (y)  obtained  by  means  of 
(XYZ)  with  the  terms  of  equation  (172)  and  its  vector  angles, 
and  uphold  the  condition  for  invariant  representation  of  (y) 
as  the  angular  displacement  proceeds.  With  this  invariance  put 
beyond  critical  doubt  such  vectors  as  (y)  take  their  place  under 
the  procedure  of  equation  (137),  and  we  have  detected  here  the 
earmarks  of  an  invariant  shift.  A  closer  superficial  agreement 
with  that  equation  results  from  the  coordination  of  derivatives 
calculated  from  equations  (170,  171),  because  the  axes  (X'Y'Z') 
remain  orthogonal  and  rotate.  With  some  watchful  avoidance  of 
confusion  in  the  notation,  the  reasoning  of  section  80  can  be 
duplicated,  and  the  result  confirmed  without  difficulty, 

Y  =^  (i'T(x')  +  j'T(y')  +  k'7(z')) 

=  [i'^,(T(.'))+r^(T(.'))+k'^^(Tc.'))] 

+  (t  X  y),     (182) 

where  (y)  in  the  vector  product  must  denote  the  shift  rate  for 
(X'Y'Z'),  and  the  rest  of  that  member  shows  the  type  of  (V(n,)). 
We  do  not  need  now  to  transcribe  the  details  of  that  develop- 
ment, with  a  less  particular  value  for  the  shift  rate. 

98.  Having  made  the  beginning  in  section  93  with  angular 
coordinate  which  may  be  placed  in  parallel  with  coordinate 
lengths,  the  above  relation  that  introduces  an  angular  velocity 


124  Fundamental  Equations  of  Dynamics 

is  liable  to  the  same  sort  of  double  reading  that  was  insisted 
upon  in  section  81,  so  that  the  change  of  reference-frame  for 
angular  velocity  would  also  come  to  the  front.  Then  using  the 
third  member  of  the  last  equation  for  illustration  of  a  more 
general  case,  its  first  group  can  be  said  to  present  angular  velocity- 
relative  to  (X'Y'Z'),  while  the  vector  product  added  transfers 
correctly  to  (XYZ)  as  a  standard.  If  this  second  branch  of  the 
idea  is  before  us,  a  continuation  of  it  in  close  likeness  to  the 
working  out  of  consequences  into  equation  (112)  suggests  itself 
naturally,  in  order  to  make  a  transfer  between  reference-frames 
that  covers  angular  acceleration,  as  the  previous  equation  pro- 
vided for  such  a  change  in  respect  to  linear  accelerations.  But 
that  general  provision  will  be  omitted,  with  the  intention  of 
considering  any  special  instance  under  its  plan  in  the  light  of 
its  own  circumstances;  and  what  attention  is  now  to  be  given  to 
angular  acceleration  will  enter  with  the  repetition  of  the  one- 
step  shift  process,  in  which  the  original  vector  (V)  is  an  angular 
velocity,  and  the  derivative  that  appears  in  particular  to  replace 
the  general  derivative  (V)  of  equation  (137)  is  an  angular  acceler- 
ation, with  the  one  standard  frame  retained,  and  no  departures 
from  invariant  values  finally  tolerated. 

That  policy  meets  the  requirements  most  frequently  made  in 
this  field,  and  indeed  the  material  that  has  grown  to  be  classic 
and  devoted  to  the  relations  of  rotation-vectors  and  their  deriva- 
tives to  dynamical  quantities,  expressed  especially  by  means  of 
Euler's  angles,  marks  its  initial  stage  at  the  point  that  we  have 
now  reached.  One  feature  of  it,  that  we  have  once  alluded  to, 
is  letting  angle  figure  as  an  algebraic  magnitude,  but  constructing 
a  sequel  where  its  two  derivatives  become  vectors,  effectively  or 
with  full  recognition.  It  cannot  be  surprising,  therefore,  that 
those  distinctions  in  respect  to  angular  quantity,  between  its 
partial  and  its  total  time-derivative,  nowhere  need  to  appear 
in  the  classic  equations;    though  we  have  been  compelled  to 


The  Main  Coordinate  Systems  125 

give  them  weight  in  the  interest  of  correct  work.  Because  both 
compensating  elements  in  equations  hke  (173)  have  their  source 
in  orientation,  a  view  that  excludes  orientation  needs  neither; 
and  the  one  magnitude  derivative  with  respect  to  time  that  is 
retained  may  within  certain  limits  raise  no  issue  whether  it  is 
partial  or  total.  There  is  however  one  place  where  comment 
has  been  the  habit  upon  something  of  defect  in  the  algebraic 
linkage,  and  where  it  is  interesting  to  discover  that  the  concept 
of  vector  angle  does  a  little  to  make  a  better  joint.  We  shall 
attempt  to  dispose  of  that  minor  matter  in  this  pause  between 
two  steps  of  the  more  important  progress. '^ 

The  comment  in  question  hinges  upon  equations  that  the 
algebraic  methods  have  always  written  equivalently  to 

.,  dt?  diA    .  . 

1  -Y  =  -T,-  cos  ^  +  TT  sm  t?  sm  tp; 


j'.^  =  _  -—  sin  ^  -f-  -r-  sm  t}  cos  <p; 
dt  dt 

*^-^  =  dT  +  dT^^^^' 


(183) 


and  where  our  sequences  of  thought  have  caused  the  substitution 
of  time  partials  everywhere  in  the  second  members.  If  we  pick 
out  one  equation  for  a  sample,  multiply  by  (dt)  and  write 

(i'-Y)dt  =  di?  cos  ^  +  d^  sin  t}  sin  (p,  (184) 

the  usual  and  perfectly  true  remark  about  it  and  its  companions 
is  to  this  effect:  The  second  members  not  being  exact  differen- 
tials under  the  ordinary  test,  because  the  equalities  are  not 
satisfied  that  would  give  for  instance 

—  (cos  <p)  =  ^  (sin  I?  sin  <p),  (185) 

>  See  Note  25. 


126  Fundamental  Equations  of  Dynamics 

there  is  some  drawback  upon  using  the  first  members.  But  if  the 
vector  plan  retains  the  total  derivatives  in  equations  (183)  and 
completes  them,  equation  (184)  becomes,  as  we  have  seen, 

(i'"j')dt  =  dt?(cos  ^  +  ^  cos  ^  sin  (p) 

+  d^^(sin  t?  sin  <p)  +  d^(—  ??  sin  ^  +  ^  sin  t?  cos  <p),  (186) 

in  which  the  coefficients  of  (dt?,  d^,  d^)  do  make  the  first  member 
an  exact  differential  by  conforming  to  the  standard  rule,  as  direct 
test  verifies.  That  particular  drawback  was  removed  by  using 
vector  angle  in  deriving  the  rotation-vector,  and  by  aiming  in 
our  calculus  deliberately  to  preserve  the  exact  differentials  that 
occurred  naturally. 

99.  For  the  kind  of  inquiry  that  comes  next  in  order,  rotation- 
vectors  in  the  standard  frame  are  an  assumed  basis  in  the  state- 
ment, being  either  given  outright  or  brought  within  reach  by 
such  data  related  to  Euler's  angles  as  the  foregoing  sections  have 
set  forth.  The  undertaking  looks  toward  expressing  angular 
acceleration-vectors  for  the  standard  frame  in  terms  of  the  same 
angles  (t{r,  d,  ^)  and  consequently  in  connection  with  some 
auxiliary  frame  like  (X'Y'Z').  In  its  main  outline  this  must 
stand  as  a  parallel  illustration  of  the  method  introduced  before; 
but  in  order  to  vary  from  mere  repetition,  let  there  be  one 
rotation-vector  (<o)  applying  to  a  rigid  body  that  is  in  pure  rota- 
tion about  the  origin  (0),  and  a  second  (y)  for  the  axes  (X'Y'Z'), 
with  whose  aid  (6)  is  to  be  determined  through  its  projections 
upon  them.  We  shall  choose  special  assumptions,  that  will  be 
found  profitable  because  they  anticipate  one  set  of  data  met  in 
real  requirements  of  investigation.  Let  that  definite  fine  of 
the  body,  which  is  to  have  the  angular  coordinates  (■^,  d)  and 
thus  specify  those  elements  of  the  body's  configuration,  always 
coincide  with  (Z');  and  to  complete  the  assignment  of  relative 
configuration  for  body  and  axes,  let  (^)  be  permanently  zero  for 
the  latter.     Therefore    (Y')   is   contained   permanently  in   the 


The  Main  Coordinate  Systems  127 

plane  (Z',  Z),  and  (X')  in  the  normal  to  that  plane.  Dis- 
tinguishing the  angles  applying  to  the  axes  as  {■ii,  O',  ^')  the 
conditions  are 

,{,'  =  ^;     ^'  =  d;        ^'  =  0;     ^  (any  value).  •      (187) 

100.  The  rotation-vectors  (w)  and  (y)  are  now  to  be  expressed, 
but  that  cannot  be  done  by  borrowing  the  forms  from  sections 
95  and  97.  For  it  is  essential  to  the  present  circumstances  that 
the  sets  of  projections  of  each  rotation-vector  must  give  that 
quantity  invariantly,  as  before  it  was  exacted  that  the  angle 
(y)  should  be  so  expressed  by  equations  (170,  171,  172).  For 
every  range  in  this  use,  equation  (134)  is  to  be  made  funda- 
mental and  characteristic.  Going  to  one  root  of  the  matter  in 
equations  (111,  116),  and  holding  to  the  leading  thought  of 
section  86,  it  becomes  formally  clear  that  no  term  like  (y  x  V) 
of  equation  (137)  can  appear  in  forms  adapted  to  the  new  inde- 
pendent start.  And  in  reason  it  is  convincing  that  projection 
at  the  moment  is  indifferent  to  past  and  future,  and  its  results 
must  be  mathematically  independent  of  a  continuing  process  to 
which  it  is  indifferent.  All  this  fits  perfectly  our  conception  of 
each  set  (X'Y'Z')  as  fixed,  and  (y)  as  a  shift  rate  among  the 
fixed  sets.  Bringing  to  equation  (173)  the  modifying  idea  that 
(y)  equal  to  zero  must  accompany  the  projection  upon  the 
individual  set  of  axes  for  the  epoch,  we  find  first  that  the  second 
groups  in  the  coefiicients  of  (ijk)  drop  away  because  they  repre- 
sent projections  of  a  term  Hke  (y  x  V),  and  secondly  that  the 
difference  between  total  and  partial  time-derivatives  disappears 
in  view  of  equations  (178,  180).  To  be  sure  this  detail  is  only  a 
roundabout  consequence  of  discarding  at  the  one  projection  that 
which  belongs  only  to  a  unified  series  of  such  projections  as  a 
whole;  but  it  has  bearing  in  dispelling  lingering  obscurities  on 
the  formal  side  of  these  matters.  The  point  would  not  need  to 
be  labored  so,  were  not  misapprehension  fostered  by  the  mis- 
nomer reference  to  moving  axes  in  speaking  of  them. 


128 


Fundamental  Equations  of  Dynamics 


This  is  preface  to  writing  the  values 

d\f^  d&  d<p 

0,  =  ttx  ^  +  O,  ^  +  ^,  ^  ; 


=  *■¥  +  *■ 


d^ 
dt 


(188) 


in  order  to  proceed  from  them  to  the  value  of  («)  that  is  con- 
nected with  the  projections  of  (o>)  on  (X'Y'Z').  It  seems  worth 
noting  that  these  may  be  corroborated  by  considering  the  par- 
ticular configuration  when  (X'Y'Z')  fall  in  (XYZ),  for  which  of 
course  equality  of  projections  must  ensue.  From  equation  (173) 
we  see  for  that  case  and  for  the  projections  of  (g>), 


rw  =  1 


d^ 
dt 


rcy)  =  0; 

i|r  =  ^  =  ^  = 


r(^) 


^Vdt  +dt; ' 


0;  and  y  =  o>. 


(189) 


It  is  true  that  the  cancellations  of  terms  arising  from  the  type 
(t  X  V)  now  follow  from  (y)  being  zero,  but  they  show  con- 
sistency in  the  final  outcome.  The  sum  in  (y(z))  is  contributed, 
part  by  turning  of  the  plane  (Y'Z)  about  (Z),  and  part  by  turning 
relatively  to  that  plane  about  (Z')  coincident  with  (Z).  Finally 
we  can  summarize  in  a  brief  rule  the  office  of  the  two  derivatives 
in  connections  like  the  present  one:  The  partial  time-derivative 
of  the  tensors  enters  where  projection  has  preceded  differentia- 
tion, and  the  total  derivative  where  differentiation  has  preceded. 
101.  By  projecting  the  rotation- vector  (o>)  upon  (X'Y'Z')  we 
find 


"(x') 


=  1' 


d^ 
dt 


=  ^1 


d^ 
dt 


(190) 


W(y')  =  3     I  dt  ^^'^'^j  ' 

,,/d^      d^  \  fd<p      d^p  \ 

"^'''  =  ^  U +dt^"^V  =  ^^VdF+dF^"^V'. 

the  tensors  being  comprehensive  or  general  values  as  explained 


The  Main  Coordinate  Systems 


129 


in  section  78,  and  therefore  open  to  differentiation,  whose  execu- 
tion yields     . 


d  d^t? 

dt  ^""'^''^  -  dt^  ' 

d   ,        ,        dV    .  dt?  drP 

d-t(-(v'))  =  dt"^sm.?  +  ^^-cos.?; 


d  , 


dV       dV 


dt?  dtA    . 


;jr'(cO(z'))    =   T.T  +  T,T  cos  t?  -  ;t-  -.,-  sin  t? 


dt 


dt2    '  dt 


dt  dt 


(191) 


The  differentiation  of  equations  (190)  needs  for  its  completion 
the  terms  introduced  by  changes  of  orientation  in  (i'j'k'),  which 


are 


^    dt?  _  diA 
dt  ~  dt 


^1  37  =  37  «:i  X  *i)  37  =  «;i  X  Oi)  ~  ; 


d^ 
dt 


d^d^ 
dt  dt  ' 


,  diA    .  /diA  ,         .„       dt?  ,         .,Ad^    . 

/  d^  y  d\l/  dt? 

=  «fi  X  JO  y~  j  sin  t?  +  (di  X  j')  ^  ^-  sin  t?; 

fdip      diA  \ 

HdF+dt^^^^j 


/di/'  ,  ,       dt?  ,  .\/d^      d<A  \ 


,/dt?d 


•P    ,   dt?d.A 
dt  +dTdI^^''^ 


(192) 


Next  resolve  the  vector  products  into  (X',  Y',  Z')  and  assemble 
the  terms  for  each  one  of  the  axes,  which  shows  for  the  results 
w  hen  reduced  by  some  cancellations 


130 


Fundamental  Equations  of  Dynamics 


.), 


(y')  =  J'  ( 


dV    .  di}  dtp      dxl^  di? 

dt^^^^^-dtdT+dtdT^"^^' 


)■ 


dV 


'^-dtdt^^^^j- 


(193) 


(194) 


In  making  the  resolution  the  components  of  the  vector  products 
to  be  used  are  shown  by 

il:i  X  *i  =  y  cos  t?  —  ^1  sin  I?;        ijfi  x  j'  =  —  *i  cos  &; 

^ixj'  =  ^i;       i|fi  X  ^1  =  ^1  sin  t?;        Oi  x  §i  =  -  j'. 

Having  obtained  by  these  operations  the  projections  of  (w)  for 
the  standard  frame  upon  (X'Y'Z'),  as  corrected  for  the  assumed 
shift  of  the  axes,  the  total  (o>)  given  by  the  vector  sum  of  the 
second  members  is  easily  seen  to  be 


d^ 
dt2 


dV 


dt2 


+  (*i  X  h) 


d^  d^ 
dt  dt 


+  ^^^^*^)dtdt    +^<f^^*^)dtdt- 


(195) 


And  this  last  form  of  the  value  for  the  angular  acceleration  of 
the  body  is  finally  to  be  compared,  on  the  one  hand  with  the 
result  of  differentiating  directly 


diA 
"  =  ^^dt  +^ 


dt?  d.^ 

^dt  +^^dt 


(196) 


and  on  the  other  hand,  with  the  standard  relation  in  equation 
(137).  The  first  of  these  comparisons  is  no  more  than  a  matter 
of  inspection,  because  the  derivatives  of  the  tensors  appear 
immediately,  and  the  known  changes  of  orientation  for  (i^i,  Oi,  ^i) 
are  exactly  accounted  for  in  the  vector  products  of  equation  (195). 
In  order  to  carry  through  the  other  comparison  we  need  for 


The  Main  Coordinate  Systems 


131 


(V(m))  the  derivatives  of  the  tensors  that  are  already  recorded 
in  equation  (191),  and  whose  vector  sum  can  be  thrown  into  the 
form,  when  the  parts  are  duly  oriented, 


^  d^   ,        dV   .       dV  ,    ,  .d'i'dt? 

^c^)  =  ■^i:^  +  ^i:i;?  +  *idt2""^  ^"^'^   '^dt  dt  * 


(197) 


dt2    '   "^'dt^ 
To  this  must  be  added 

N       /      d^  dt?\     /      diA  dt?  d<s\      ,_„, 

whose  expansion  reduces  to 

dxl/  d<a  dd  d<p 

Ctx..)  =  «.x4,)di^df+(».xWgfdf,       (199) 

and  confirms  through  the  sum  of  equations  (197,  199)  the  former 
value  of  (o)).  Notice  the  difference  in  the  segregation  for  the 
two  groupings,  by  which  the  same  term  can  be  attributed  at  will 
to  change  of  direction  or  of  magnitude. 

The  components  of  (co)  in  (XYZ)  are  obtainable  in  the  forms 


<0(x)    =    1 


dd  d<p   .         .       \ 

cos  ;/'  +  37  sm  t?  sin  i/-  I  ; 


"(y)  = 


dt 

d^ 

dt 


sin  \l/ 


dt 

d^ 

d^ 


sin  d  cos  \l/ 


): 


(200) 


through  which  another  plain  road  is  opened  to  determine  (<!>); 
but  we  shall  not  go  further  here  than  to  indicate  it. 

Polar  Coordinates. 

102.  The  system  known  as  polar  coordinates  is  a  fitting  sequel 
to  what  has  just  been  done,  because  Euler's  angles  that  we  have 
denoted  by  (i^,  -6^)  are  universally  employed  to  orient  the  radius- 
vector  (r)  whose  pole  is  then  taken  at  our  origin  (0) .     The  angle 


132  Fundamental  Equations  of  Dynamics 

(^)  is  obviously  superfluous  when  we  are  concerned  with  one 
hne  only  and  not  with  a  body,  even  when  (r)  moves  in  three 
dimensions;  and  when  a  limitation  to  the  uniplanar  conditions 
is  imposed  the  pole  is  most  often  located  in  the  plane  of  motion, 
and  then  of  the  three  angles  (tl/)  alone  needs  to  be  used.  We 
shall  guide  the  development  toward  the  relations  for  three 
dimensions,  and  afterwards  call  attention  to  some  briefer  state- 
ments for  the  uniplanar  case. 

If  we  write  the  radius-vector  (r)  as  the  product  of  its  unit- 
vector  (rO  and  its  tensor  (r),  according  to  one  normal  scheme  of 
the  vector  algebra,  the  time-derivative  (t)  takes  on  the  form 

f  =  ri^+rir,  (201) 

with  unforced  separation  of  the  entire  directional  change  from 
that  which  refers  to  the  algebraic  magnitude.  By  means  of  the 
results  now  at  our  disposal,  the  vector  (y)  in  application  to  the 
single  line  (r)  would  lead  straight  to  the  expression  for  the 
velocity  of  (Q)  at  the  extremity  of  (r), 

dr       ,         ,  dr       ,  ^    dd/       ,  x    dt?        , ^ 

V  =  ri^-t-  (rxr)  =  ri^  +  (<rixrOr^  +  (dixri)r^-  .     (202) 

From  the  second  member,  we  infer  at  sight  the  truth  of  one  usual 
statement  about  (v) :  That  it  includes  simultaneous  motion  on  a 
sphere  centered  at  the  pole  of  (r),  and  growth  of  (r)  in  length. 
So  long  as  we  think  strictly  in  the  terms  indicated,  there  is  no 
rotation  according  to  our  use  of  that  word;  we  deal  with  (y) 
merely  as  the  angular  velocity  of  the  one  line.  But  when  the 
third  member  of  the  last  equation  is  drawn  in,  the  set  of  axes 
(X'Y'Z')  as  laid  down  in  section  93  reappears,  since  the  three 
parts  of  the  velocity  constitute  always  an  orthogonal  set,  of 
which  (r)  itself  would  be  (Z')  in  our  adopted  convention,  coin- 
ciding with  (Z)  for  zero  values  of  (i|r,  ^).     The  completed  con- 


The  Main  Coordinate  Systems  133 

sistent  identification  of  axes  and  their  true  rotation-vector  gives 

'^'^^^dt~'~*^dt'        ^  =  0  permanently. 


(203) 


It  is  self-evident  that  these  three  projections  are  an  invariant 
equivalent  for  (v),  because  they  are  in  their  source  only  the 
three  parts  of  (t)  in  the  standard  frame.  But  we  can  also  repeat 
the  remark  attached  to  equation  (138),  and  enlarge  it  in  the 
direction  of  presenting  these  polar  coordinate  relations  for  velocity 
in  the  light  of  a  narrowly  specialized  instance  within  more  elastic 
conditions. 

Instead  of  binding  (X'Y'Z')  to  coincidence  of  (Z')  and  (r), 
let  the  axes  rather  move  about  the  origin  (O)  as  allowed  by  any 
general  value  of  the  rotation-vector  (y).  The  configuration  of 
(r)  in  the  frame  (X'Y'Z')  will  be  shown  generally  by 

r  =  i'x'  +  j'y'  +  k'z';  (204) 

and  for  those  suppositions  the  general  values  of  (V)  and  (V(ra)) 
in  equation  (137)  will  assume  the  form 

dr  ,    ,  d)/'       ,  ,    dt> 

r  =  r.-+(,I,.xrOr^  +  (*:xrOr^ 

.,dx'       ..dy'       ,  ,dz'       ,         ,        ,^^^x 

The  effect  of  that  particular  choice  for  the  rotation-vector  in 
equation  (203)  is  then  put  clearly  in  evidence:  the  velocity  of 
(Q)  at  the  extremity  of  (r),  but  reckoned  relatively  to  the  frame 
(X'Y'Z'),  is  thrown  exclusively  upon  the  axis  (Z'),  while  (x',  y') 
remain  permanently  at  zero,  and  the  term  (y  x  r)  is  left  to  bring 


134 


Fundamental  Equations  of  Dynamics 


in  all  of  both  components  that  (v)  shows  parallel  to  (X')  and 
to  ( Y') .  Or  in  the  alternative  reading,  the  correction  for  shift  of 
orientation  being  perpendicular  to  (r),  it  is  segregated  com- 
pletely from  the  only  change  in  tensor  magnitude  that  is  allowed 
to  become  realized  in  (X'Y'Z'). 

103.  The  natural  order  proceeds  next  to  take  up,  with  polar 
variables  as  instruments,  the  task  of  expressing  the  polar  com- 
ponents of  the  acceleration  with  which  (Q)  moves  relatively  to 
the  standard  frame,  and  which  can  be  determined  otherwise, 
as  we  know,  by  projecting  the  resultant  (t)  upon  the  directions 
of  (X'Y'Z')  at  the  epoch.  However  these  projections  may  be 
written  originally,  the  translation  into  functions  of  (r,  ijr,  d)  is  a 
matter  of  algebra  only.  Leaving  that  method  aside,  the  details 
will  be  worked  out  in  two  ways,  both  moving  with  reasonable 
directness  toward  the  end  in  view,  and  each  having  its  own 
interest  through  the  vector  algebra  of  it.  Let  us  carry  out 
first  the  application  of  equation  (137).     It  gives 


(m) 


V    dt2  ^dt  dt/ 


+  3 


■( 


dV    .  dr  di/'  .  diA  di? 

^dt"^^^^^  +  dtdF^^"^  +  d^dt 


(y  X  v)  = 


r      d^  di?  1 


+  k'    3r, 


r     dr       ,  s   d^A       ,  ,   d§ 

I  ri^  +  (tti  X  Ti)r^  +  (di  X  ri)r 


}l 


(206) 


dt    '    '"^'^'^'^dt 

As  a  help  in  expanding  the  second  equation  these  relations  enter: 

(^1  X  rO  =  ■»!  sin  ??; 
iti  X  (i^i  X  Ti)  =  —  i'  sin  ^  cos  ^  —  Ti  sin^  ?>; 
1^1  X  (*i  X  Ti)  =  Oi  cos  i};        (^1  X  ri)  =  i'; 
^1  X  (tti  X  ri)  =  0;        di  X  (^i  x  ri)  =  -  ri. 


(207) 


The  Main  Coordinate  Systems 


135 


Summing  the  items  in  their  proper  orientation,  the  polar  com- 
ponents of  (v)  are  found  to  be 

d^t?      „drdt?         /di/'V  • 
dt2  ^     dt  dt 


(208) 


.   /   d^t?      „drdt?         /di/'V  •  \ 

^(x')  =  1  (^r^  +  2^t  dt  -  'U  J  ''°  ^  '"'V' 

.,  /    dV  .         .   ^d^dr  .  „  diJd./'  \ 

^(y')  =  J'  (^^dt^si^'^  +  2d^dt'^°'^  +  ^^dTdT^^'^  j' 

^-'>=^(dt^-Kdr)-<dt)«-^^)- 

The  second  development  picks  up  its  thread  at  equation  (201), 

and  differentiates  that  again  as  it  stands;  so  the  first  stage  shows 

immediately 

d^r      _.  dr 


^^^  =  '^dti+2f:^^+r-:r; 

and  carrying  out  some  of  the  indicated  operations  yields 

fi  =  (yxTi); 

i-'i  =  (y  X  ri)  +  (y  X  fi)  =  (y  X  ri)  +  (y  x  (y  x  ii)); 

:   djP   ,    .   dV  ,    A   dt?   .    .    d^t? 
'   dt        * 


Y  =  <:i:Tr  +  ^idt'^  +  ^idt+^^dt2' 


d^ 
with  1^1  constant,  ^i  =  tt  (^i  x  ^i); 

^dV  .   /,  ,dxlydr}\ 

(y  X  rx)  =  (iti  X  ri)  ^  +  (^  (.<li  X  ^i)  ^^^- j  X  ri 


Y  X  (y  X  Ii)  =  [iti  X  (tti  X  Ti)]  (  -7- 


m 


d2|? 
+  (^ixri)^; 


+  [itix(0,xrO]^^+[d:x«:.xrO]^^ 


+  [^1  y  (^ 


10 


•^'^^K^J 


(209) 


(210) 


136  Fundamental  Equations  of  Dynamics 

Substituting  these  values  in  equation  (209),  it  is  recognizable 
readily  with  the  aid  of  equations  (207)  that  the  results  of  the  two 
methods  are  in  perfect  agreement. 

104.  The  adjustment  of  the  foregoing  analysis  to  the  simplified 
conditions  of  uniplanar  motion,  where  the  pole  for  (r)  is  taken 
in  the  plane  of  the  motion,  will  make  (•&)  constantly  a  right  angle, 
so  that  (r)  revolves  in  the  equatorial  plane  of  the  sphere  whose 
polar  axis  is  (Z).  In  adaptation  to  that  case  the  velocity  com- 
ponents are 

V(z')  =  V(r)  =  rx^;  V(.v')  =  i  yr^  J;  V(x')  =  0;  (211) 
and  the  acceleration  components  become 


V(z')     =    V(r) 


/d^r         fdxPY\ 

dt2  "^   dt  dt  y ' 


(212) 


V(x')  =  0. 

Even  on  this  simpler  level,  and  after  removing  those  complica- 
tions which  belong  to  the  freedom  in  three  dimensions,  the 
same  feature  remains  prominent  through  all  the  results;  in  one 
sense  the  idea  of  superposition  fails.  For  though  the  resultant 
velocity  contains  neither  more  nor  less  than  the  parts  due  to  the 
radial  motion  by  itself  and  the  revolution  by  itself,  we  cannot 
build  up  in  that  fashion  the  acceleration  (t)  of  equations 
(208),  nor  yet  of  equations  (212).  In  the  latter,  the  second  term 
in  the  coefficient  of  (j')  does  not  belong  to  the  radial  motion,  nor 
to  the  circular  motion,  but  it  appears  only  when  these  two  types 
coexist.  And  under  the  broader  conditions,  the  coexistence  in 
pairs  of  the  three  component  velocities  asserts  itself  through 
the  terms  in  the  acceleration : 

^'my^  y(4ty.  y{-m-  <-) 


The  Main  Coordinate  Systems  137 

In  view  of  their  obtrusive  symmetry,  it  is  somewhat  surprising 
that  the  force  depending  on  the  third  of  the  group  should  have 
invited  and  fixed  nearly  exclusive  attention:  it  is  the  famous 
compound  centrifugal  force  with  which  the  name  of  Coriolis  has 
been  associated.^ 

Approaching  along  the  line  now  laid  down  to  follow,  these 
terms  can  be  traced  intelligently  to  a  common  origin  in  the 
nature  of  the  coordinate  system  that  is  being  employed;  their 
appearance  is  connected  essentially  with  the  changes  of  direction 
pecuhar  to  the  descriptive  vectors  that  are  used.  On  that  side, 
the  parts  of  the  force  that  match  such  accelerations  may  be 
declared  mathematical,  though  it  must  be  granted  that  they  can 
become  sound  physics  too,  whenever  those  descriptive  vectors 
are  closely  fitted  to  the  physical  action.  In  a  centrifugal  pump, 
a  force  that  goes  with  the  coefficient  of  (j')  above  does  work 
and  strains  the  structural  parts.  But  the  same  term  shows  in 
the  algebra,  when  constant  velocity  is  referred  to  a  pole  lying 
outside  the  straight  line  path,  although  no  net  force  at  all  can 
then  be  active.  It  is  also  a  significant  fact  that  the  factor  (2) 
in  each  case  makes  its  appearance  because  two  terms  coalesce, 
whose  function  is  different  in  respect  to  the  vector  quantities 
that  they  affect.  It  is  half-and-half  change  of  magnitude  in 
one  vector  and  change  of  direction  in  a  second  distinct  vector, 
as  our  process  of  derivation  demonstrates.  So  the  force  of 
Coriolis  cannot  give  a  definitive  account  of  gyroscopic  phenomena 
on  the  basis  of  an  incident  in  the  algebra;  first,  it  must  be 
exhibited  to  correspond  with  traceable  dynamical  action.  The 
same  lesson  is  enforced  here  as  by  the  matters  broached  in 
sections  35  and  57,  of  which  the  latter  is  peculiarly  pertinent  in 
that  it  brings  forward  the  idea  that  angular  acceleration,  and 
therefore  the  coexistence  of  rotations  about  (t{ri)  and  (^i)  that  is 
characteristic  of  the  compound  centrifugal  force,  may  come  about 

>  See  Note  26. 


138  Fundamental  Equations  of  Dynamics 

in  the  absence  of  all  force-moment,  as  a  symptom  that  control 
is  absent,  not  that  it  is  present  and  is  producing  these  effects. 
105.  The  general  values  of  equation  (208)  cover  as  a  special 
case,  it  is  plain,  the  condition  that  (r)  shall  be  constant  in  length 
which  goes  with  a  pure  rotation  about  (O).  Consequently  if 
we  make  that  assumption  here,  the  special  value  of  (♦)  that  is 
obtained  must  be  reconcilable  with  the  determination  made  in 
sections  54  and  101.  Only  the  latter,  in  its  turn,  must  be  special- 
ized for  a  point  situated  in  its  axis  of  (§i),  which  is  now  also  that 
of  (ri).  The  notation  in  the  two  sections  is  consistent  with  the 
same  supposition  about  the  rotation- vector  (y)  of  (X'Y'Z'); 
and  the  axis  (Z')  is  common  to  both  inquiries.  But  it  will  be 
observed  that  (^i)  of  section  (101)  is  identified  with  (i'),  and  (di) 
of  equations  (203)  is  paired  with  (j');  and  hence  a  comparison 
of  results  must  adopt  in  correspondence 

(i');  (JO;         (k');         [Equations  (193)] 

(JO;         (-i');         (kO;         [Equations  (2O8)] 

in  order  to  preserve  the  right-handed  cycle. 

If  (r)  is  constant  in  length  the  terms  remaining  in  equations 
(208)  are 

V(x')=i'(^r^-r(^-j^j  sm^cos^j; 


^(y''"   = 


(214) 


.,  /    dV    .  ^    dip  dd  \ 

)=3   (^r^sm^  +  2r^^cos^j; 

And  the  vector  sum  of  these  must  agree  with  equation  (72)  after 
the  latter  has  been  adapted  to  the  point 

z'  =  r;         x'  =  y'  =  0.  (215) 

We  have  for  use  with  equations  (72,  188,  193) 


(216) 


The  Main  Coordinate  Systems  139 

(fc)  X  r)  =  i'(«(y')r)  -  j'(^(x')r); 
<o(fa)-r)  —  r(a)*) 

r      dxP  d^  d<pl/'    fdrP  d<p\\ 

When  the  multiphcations  are  carried  out  and  the  items  duly- 
oriented  by  the  plan  explicitly  recognized  for  equations  (214), 
the  values  are  found  in  agreement  at  all  points. 

The  special  circumstances  to  which  equations  (214)  conform, 
make  them  express  the  acceleration  of  a  point  in  the  symmetry- 
axis  of  a  top  or  gyroscope  when  it  is  spinning  about  that  axis 
while  the  latter  is  executing  any  motions  that  change  (d)  and 
(it).  Beside  the  utility  of  this  value  in  application  to  the  problem 
of  the  top,  and  the  consolidation  that  the  conclusions  attain 
through  the  comparison,  it  is  particularly  instructive  to  follow 
carefully  and  in  detail  the  appearance  of  terms  in  the  acceleration, 
and  their  various  disappearances  by  cancellation.  Then  one 
learns  to  cross-examine  the  mathematics  and  to  discount  sensibly 
its  evidence  or  suggestion  as  to  just  what  dynamical  processes 
are  in  operation. 

106.  The  fact  that  the  resolution  into  polar  component  shapes 
itself  in  accommodation  to  each  individual  radius-vector  prevents 
the  introduction  of  any  usefully  general  integrations  to  include 
extended  masses.  As  a  substitute  recourse  is  had,  where  the 
radius-vector  enters  naturally,  to  plans  like  that  worked  out 
for  the  rotation  of  a  rigid  body,  which  has  contrived  to  extract 
the  common  elements  (to)  and  (ci)  for  use  with  all  radius-vectors, 
and  the  moments  of  inertia  as  factors  that  cover  the  whole  mass. 
The  polar  components  that  have  been  deduced  are  then  limited 
practically  to  one  mass-element  or  to  the  particle  at  the  center  of 


140 


Fundamental  Equations  of  Dynamics 


mass  of  the  body.  For  the  latter  case,  there  is  no  difficulty  in 
writing  down  for  the  six  fundamental  quantities  the  parts  of 
their  standard  frame  values  that  match  the  orthogonal  polar 
projections.     These  are: 


drl 
^^d-tj 


(217) 


Q  =  m  1^  (di  X  rOr  —-  +  {^,  x  ri)r  ^  + 

=  Q(x')  +  Q(y')  +  Q(z'); 
E  =  >[v2(.')  +  v2(/)  +  v2(.')] 

=  E(s')  +  E(y')  +  E(z'); 
H  =  m[-i'(rv(,'))+j'(rv(.'))] 

=  H(x')  +  H(y');     [H-iz')  =  0]; 
R  =  m[i'v(^')  +  j'v^y'-)  +  kV(^')] 

=  R(x')+  R(y')  +  R(z'); 
P  =  R(x')V(x')  +  R(y')V(y')  +  R(z')V(z'); 
M  =  -  i'(R(y'))  +  j'(rR(,'))  =  M(.')  +  M(y'); 

[M(.')  =  0].  J 

As  an  addendum  to  the  separation  of  power  or  activity  (P)  into 
its  parts  it  is  worth  noting  that  the  total  force  corresponding  to 
the  heading  (y  x  v)  of  equation  (206)  can  finally  contribute 
nothing  to  the  work  done.  It  must  of  necessity  be  perpendicular 
to  (v)  and  therefore  ineffective  in  the  product  (R-v).  Amounts 
of  work  per  second  may  be  yielded  in  the  parts  of  (P)  by  the 
inclusion  of  these  directional  forces,  but  they  must  be  self-com- 
pensating and  give  zero  of  work  in  the  aggregate.  Their  behavior 
in  both  respects  toward  power  is  similar  to  that  of  normal  force 
that  is  confined  to  changing  direction  in  resultant  momentum. 
Under  (V(m)),  other  elements  of  force  may  be  entered  that  also 
give  change  of  direction  to  (mv) ;  this  function  it  may  share  with 
(y  x  v).     But  (V(m))  has  monopoly,  as  was  pointed  out  earlier, 


The  Main  Coordinate  Systems  141 

of  bringing  about  all  changes  of  magnitude  in  (v),  and  hence  in 
(mv).  It  is  plain  common  sense  to  confirm  these  conclusions  by 
the  observation  that  what  happens  to  coordinates  merely — to 
the  descriptive  vectors  as  we  have  called  them — cannot  affect 
the  physical  data  that  they  are  devised  to  describe. 

Hansen's  Ideal  Coordinates. 

107.  By  the  trend  of  the  standard  illustrations,  it  cannot  fail 
to  have  grown  conspicuous  already,  how  varied  the  available 
combinations  must  be  and  how  many  kinds  of  adjustment  to 
special  purposes  are  rendered  possible,  when  once  such  resources 
and  expedients  have  been  brought  under  fair  control,  and  a 
definite  formulation  of  the  ends  sought  has  been  arrived  at. 
The  next  instance  in  order,  the  ideal  coordinates  so  named  by 
Hansen  who  proposed  them,  is  adapted  to  strengthen  that  per- 
ception.^ The  invention  of  the  plan  seems  to  have  been  con- 
sciously directed  by  a  purpose,  and  it  finds  a  place  here  because 
it  has  made  its  standing  good  for  certain  fields  of  application. 
As  would  be  natural  to  surmise,  the  proposals  that  have  won 
acceptance  have  been  gleaned  by  the  sifting  of  actual  and  con- 
tinued trial  among  the  larger  number  submitted  for  general 
approval.  Ideal  coordinates  are  made  to  follow  upon  the  polar 
system  here  because  the  radius-vector  still  remains  a  prominent 
element  in  their  specifications;  and  on  this  account,  they  too 
have  no  immediate  range  beyond  tracing  the  motion  of  one 
particle  or  mass-element.  It  will  be  recognized  that  they  pursue, 
like  the  other  coordinate  systems  that  have  been  discussed,  the 
object  of  stating  standard  frame  values,  but  in  more  elastic 
partition  of  the  totals  than  (XYZ)  itself  can  furnish. 

The  chief  concern  of  ideal  coordinates  is  with  velocity,  and  its 
main  course  may  be  called  a  response  to  the  question,  in  what 
direction  can  the  restrictions  upon  the  frame  (X'Y'Z'),  that  the 

1  See  Note  27. 


142  Fundamental  Equations  of  Dynamics 

polar  system  has  been  seen  to  impose,  be  loosened  without  im- 
pairing the  invariance  of  (v)  that  the  polar  components  retain. 
That  point  being  secured,  the  other  consequences  entailed  are 
left  in  whatever  form  they  may  happen  to  appear.  In  this  way 
it  becomes  part  of  the  inquiry  to  ascertain  how  the  expression  of 
acceleration  is  affected  by  the  assumed  conditions.  The  frames 
(X'Y'Z')  and  (XYZ)  continue  with  a  common  origin  (O). 

108.  If  we  add  to  the  suppositions  of  section  102  a  rotation  of 
(X'Y'Z')  about  (Z')  that  can  be  of  any  assigned  magnitude, 
equation  (202)  will  be  written,  when  as  before  we  identify  (^i, 
k',  and  rO, 

dr  dJ/  d«?  d(p 

V  =  ri  ^  +  (»l:i  X  ri)r  ^  +  (*i  x  ri)r  ^  +  (4i  x  r^r  ^[  ;  (218) 

but  the  difference  introduced  is  only  formal  since  (^i,  Ti)  are 
identical  unit-vectors,  and  in  this  frame  (X'Y'Z')  it  is  still  the 
coordinate  (r)  or  (z')  alone  that  can  differ  from  zero,  while  the 
same  corrections  make  the  previous  invariant  representation  of 
(v)  persist.  This  puts  before  us  the  nucleus  of  Hansen's  idea,  as 
vector  algebra  allows  us  to  condense  it.  Now  it  will  not  be 
overlooked  that  (V(x'),  V(y'),  V(z'))  as  determined  by  equation 
(203)  are  the  components  of  (v)  in  that  frame  of  permanent 
configuration  in  (XYZ)  for  which,  with  (§)  equal  to  zero,  the  frame 
(X'Y'Z')  is  the  indicator  at  the  epoch.  But  it  follows  from  the 
form  of  equation  (218)  that  a  whole  group  of  fixed  frames  which 
at  the  epoch  have  (Z')  in  common  and  are  distributed  through 
all  azimuths  round  that  axis  for  the  range  (0,  27r)  in  (<p),  satisfy 
first  the  relation  for  the  vector  sum 

di/'  dt} 

V(x')  +  V(y')  =  (t|fi  X  ri)r  ^  +  (*i  X  rOr  ^-  ,         (219) 

and  accordingly  for  the  invariance  of 

V  =  V(.')  +  V(/)  -f  V(.').  (220) 


The  Main  Coordinate  Systems  143 

Whatever  the  direction  therefore,  in  which  the  extremity  of  (r) 
is  instantaneously  moving  parallel  to  the  (X'Y')  plane,  it  is 
possible  to  select  at  that  epoch  among  the  group  mentioned 
above  one  frame  for  which  (V(x'))  is  zero,  and  another  for  which 
(V(y'))  is  zero;  and  whichever  alternative  is  chosen  of  these  two 
it  is  further  open  to  attempt  determining  the  rate  of  the  rotation 
about  (Z')  so  that  this  one  component  remains  permanently  zero. 
We  shall  return  presently  and  develop  consequences  of  those 
possibilities,  after  pausing  to  insist  a  little  upon  equation  (220) 
which  has  not  yet  been  particularized  in  that  sense. 

109.  In  order  to  come  nearer  to  the  form  of  statement  that 
Hansen  was  compelled  to  employ,  go  back  to  section  89,  where 
equations  (150,  151)  express  the  invariance  of  (r)  in  frames 
having  a  common  origin.  Let  us  pass  on  to  consider  equations 
(154),  noticing  how  the  added  invariance  of  (v)  necessitates  the 
vanishing  of  the  last  group  of  terms  in  each  of  them,  for  which 
one  condition  extracted  from  equation  (162)  is  seen  to  be  that  (y) 
though  differing  from  zero  is  colinear  with  (r).  For  our  benefit 
just  now,  this  signifies  that  if  two  frames  give  equivalent  sets  of 
components  for  the  same  resultant  velocity,  the  equivalence  will 
not  be  disturbed  by  allowing  one  of  them  to  be  subject  to  a  shift, 
provided  that  the  axis  of  it  lies  in  the  radius-vector  at  the  epoch. 
Then,  as  Hansen  puts  it,  equations  (151,  154)  will  exhibit  the 
same  type  in  their  forms,  with  velocities  replacing  everywhere 
the  corresponding  coordinates,  and  the  ideal  for  (x'y'z')  has  been 
reached.  As  we  have  approached  it  there  are  two  stages:  the 
shift  of  (X'Y'Z')  in  the  angular  coordinates  (t{f,  *)  is  not  without 
influence  upon  the  relations,  but  it  has  been  compensated  in 
equations  (203),  and  adding  then  a  supplementary  shift  about 
(ti)  that  is  also  (^0  leaves  this  compensation  untouched. 

The  zero  value  of  (^)  having  been  standardized  for  equation 
(203)  with  (X')  in  the  plane  (Z',  Z),  for  the  more  general  value 
of  (^)  that  is  now  contemplated  we  should  write 


144  Fundamental  Equations  of  Dynamics 

.,  /    dt?  .      d^    .         .       \ 

V(x')  =  1   (  r-rr  cos  ^ •+  r-r-'  sin  t?  sin  v?  I ; 

(221) 
.,  /         dt?    .  ,      d^    .  \ 

V(y')  =  ]   I   -  r  ^  Sin  ^  +  r  ^  Sin  t?  COS  ^  I . 

And  if  we  settle  upon  making  (V(y'))  zero,  the  proper  value  of  (^) 
at  the  epoch  is  determined  by 

di/'    . 
r  ^  sin  ?? 

tg  <p'  =         ^^      .  (222) 

^dt- 

Let  us  retain  (y)  for  the  rotation-vector  of  (X'Y'Z'),  and  dis- 
tinguish by  (to)  the  angular  velocity  of  (r),  so  that  in  the  subse- 
quent details 

dip  d??  d<p  d\p  dd^       ,     „, 

Then  under  the  condition  of  adjustment  shown  by  equation  (222) 
we  have 

v,.,  =  (oxr)=i'[r^(^y+r^(^ysin^^J; 

(224) 
dr 

V(y')  =  0;         V(,')  =  fi^- 

110.  The  execution  of  this  manoeuvre  reduces  the  statement, 
so  far  as  velocities  are  concerned,  to  one  of  motion  in  an  instan- 
taneously oriented  plane  (Z'X'),  with  a  resolution  of  (v)  for  the 
standard  frame  along  the  radius-vector  and  the  perpendicular  to 
it  in  that  plane.  The  values  of  the  components  conform  per- 
fectly in  type  to  those  of  the  similar  projections  in  the  permanent 
plane  of  uniplanar  conditions;  and  the  prospect  is  opened  for 
success  in  determining  such  a  rate  of  rotation  about  (r)  as  will 
perpetuate  the  instantaneous  relations  in  exactly  this  form 
when  they  have  been  established  at  some  one  epoch;  this  involves 


The  Main  Coordinate  Systems 


145 


keeping  the  values  of  (V(y'))  continuously  at  zero,  though  it  is 
always  reckoned  in  the  normal  to  the  shifting  plane  (Z'X')- 
The  examination  of  the  arrangement  requisite  to  that  end  is 
connected  with  the  question  about  components  of  the  accelera- 
tion (t),  and  we  shall  make  our  beginning  there. 

Recorded  in  equations  (208)  are  the  projections  of  (t)  for 
(XYZ)  upon  the  (X'Y'Z')  axes  as  located  by  (^  =  0) ;  and  from 
them  can  be  calculated  the  equivalent  set  of  projections  upon 
the  axes  (X'Y'Z')  located  by  the  general  value  of  (^),  precisely 
as  equation  (221)  does  this  for  velocit3^  Those  projections  can 
finally  be  particularized  for  the  angle  (^')  assigned  by  equation 
(222)  to  satisfy  its  announced  condition.  Distinguish  the  last 
named  components  of  (t)  temporarily  as  (v'(x"),  t(y"),  ^(z")); 
they  are  given  by 

*-(x")  =  i"(v(x')  cos  <p'  +  V(y')  sin  ^'); 

^(y")  =  j"(-  V(x')  sin  <p'  +  V(.v')  cos  <p'); 

t(,")  =  k"(v(,'));    with  k"  =  k'  =  ri, 

the  new  unit-vectors  being  (i"j"k")- 


(225) 


In  the  text  of  section  103,  the  components  of  (v)  happen  to  find 
expression  through  polar  variables,  but  that  is  plainly  only  an 
incident  of  the  sequence  in  which  they  were  developed;  they 
might  just  as  well  have  been  derived  from 

d^x 
dt2 


^(x")     = 


^y' 


)  =i"(i" 

)  =  j"(r- 

)  =  k"  (k" 


i3I^+i' 


■^dt^+^ 


d^\ 

dtV' 


+  3' 


d^ 

dt^ 

•  dx2      ,  , 
^dt^+^ 


^+r.k^V 
dt^^^   ^dtV' 

dtV  '  J 


•^dl^+^" 


(226) 


or  in  some  other  equivalent  fashion,  the  choice  depending  upon 
how  the  data  are  presented.     It  is  another  consequence  of  this 


146  Fundamental  Equations  of  Dynamics 

idea,  that  the  original  shift  of  (X'Y'Z')  in  (ijf,  d)  belonging  to 
polar  components  is  unessential;  in  effect  it  drops  out  of  con- 
sideration through  the  allowances  for  its  presence  when  equations 
(208)  were  made  correct,  as  we  saw  also  in  speaking  about 
equations  (203).  The  vital  element  in  these  ideal  coordinates  is 
the  accompanying  rotation  about  (r)  which  has  been  relied  on 
at  critical  points  to  secure  at  once  invariance  and  simplification 
in  the  relations  for  the  velocities,  and  whose  consistent  intro- 
duction into  those  for  acceleration  we  are  now  prepared  to  finish. 
In  order  to  accentuate  the  real  dissociation  from  the  polar 
scheme,  let  us  think  definitely  in  the  terms  suggested  for  equations 
(226),  of  two  coincident  frames  in  the  configuration  with  (XYZ) 
designated  by  (tjf,  ^,  ^'),  of  which  one  is  fixed,  while  the  other  is 
departing  from  coincidence  by  rotation  about  the  (r)  of  the  epoch. 
We  will  temporarily  call  the  rate  of  this  departure  (u)  in  substi- 
tution for  the  time  rate  (^i  d^j/dt). 

111.  Then  the  specialization  of  equation  (137)  to  these  cir- 
cumstances gives,  if  we  particularize  the  velocities  also  as  (V(x"), 
V(y"),  V(z")),  and  remember 

U(,")  =  u;        U(^")  =  U(y")  =  0;        V(y")  =  0;      (227) 
^(x")  =  i"^(v(x")); 


^(y")  =  J 


r[^(v(y"))  +  (ua,r)]; 


'('") 


=  k"f^(v,0). 


(228) 


Hence,  in  order  that  these  values  may  be  reconciled  with  a 
permanent  zero  value  for  (V(y")),  the  magnitude  of  (u)  niust  be 
adjusted  to  the  acceleration  parallel  to  (j")  of  the  epoch,  which 
for  the  present  purpose  we  may  suppose  to  be  one  among  the 
data,  as  well  as  the  velocity  component  (i"(uT)).     At  the  same 


The  Main  Coordinate  Systems  147 

time,  as  the  forms  of  the  last  equation  show,  the  accelerations 
parallel  to  (k",  i")  are  reckoned  as  though  those  were  constant 
unit-vectors.  But  it  is  plain  that  the  existence  of  shift  cannot 
disappear  completely  from  acceleration  and  from  velocity  too, 
because  the  necessary  conditions 

(u  X  r)  =  0;         (u  X  v)  =  0;        with  (u)  not  zero;    (229) 

are  incompatible,  so  long  as  (v)  and  (r)  are  not  parallel. 

There  is  a  strong  natural  suggestion,  through  the  connection 
and  the  form  in  which  these  ideal  coordinates  have  come  to  our 
attention,  that  they  bear  by  their  intention  upon  the  astronomical 
problems  that  occupy  themselves  with  orbits  whose  differential 
sectors  are  drawn  out  of  one  containing  plane  by  disturbing  forces. 
To  this  conception  of  a  continuous  succession  of  osculating  orbits 
the  method  is  ingeniously  accommodated,  with  a  separation 
that  is  of  practical  advantage  between  the  forces  (mV(x")), 
(m^(8")),  whihc,  as  it  were,  control  the  orbit-element  of  the 
epoch,  and  the  force  (mt(y"))  into  which  the  distorting  influence 
is  collected.  Yet  interest  in  the  method  should  not  be  confined 
to  astronomers,  because  its  device  is  repeated  with  only  the 
modifications  that  the  new  conditions  impose  under  the  next 
heading,  when  the  osculating  circle  of  curvature  is  brought  into 
relation  with  any  curved  path  of  a  moving  point;  and  the 
parallelism  is  an  instructive  feature  for  our  discussion. 

Resolution  on  Tangent  and  Normal. 

112.  The  local  tangent  and  normal  to  the  path  of  a  moving 
point  afford  a  coordinate  system  that  has  been  in  general  use 
since  the  days  of  Euler,  but  its  employment  for  velocities  could 
not  be  carried  beyond  the  rudimentary  stage  of  indicating  the 
set  of  values  (0,  0,  v)  in  every  such  application.  It  is  clear  that 
this  remark  includes  with  equal  force  momentum  and  kinetic 
energy  that  contain  no  other  kinematical  factor  than  velocity. 


148  Fundamental  Equations  of  Dynamics 

The  resolution  tangentially  and  normally  has  that  ground  for 
concerning  itself  solely  with  acceleration  and  with  dependent 
dynamical  quantities  like  force,  power  and  work.  In  this  it 
differs  from  the  coordinate  systems  that  have  been  occupying 
us  hitherto:  by  not  being  serviceable  in  more  than  one  stage  of 
differentiation,  whereas  the  terms  of  the  other  systems  have 
linked  with  two  derivatives  at  least.  How  the  tangent-normal 
plan  branches  off  from  the  radius-vector  series  appears  when  we 
write 

dr 
r  =  Tir;        f  =  v  =  f ir  +  ri  ^  ; 

(230) 
dv 
r  ^  V  =  v,v  -F  vx  ^  ; 

and  compare  the  last  equality,  that  realizes  the  separation  along 
tangent  and  normal  to  the  path,  with  equation  (209)  that  con- 
tinues the  polar  component  scheme.  Because  one  stage  does 
isolate  itself  thus,  it  becomes  feasible  for  it  to  remain  bound  by 
the  invariance  test  for  a  quantity  with  which  it  connects,  and 
yet  take  on  the  quality  of  a  mixed  plan  in  other  respects.  A 
plan  mixed  or  composite  in  regard  to  the  standard  frame,  by 
dealing  with  comparison-frames  (O',  X'Y'Z')  whose  (r')  and  (f') 
are  not  invariant  with  (r)  and  (f),  though  (r")  and  (f)  are  thus 
related.  Section  77  furnishes  all  needed  reminder  about  like 
combinations. 

Such  realities  as  the  exclusion  of  normal  force  from  effect 
upon  power  have  thrown  tangential  force  into  stronger  relief; 
and  the  more  impressive  function  of  the  latter  in  changing  mag- 
nitudes. Some  plan  or  other  of  resolution  for  acceleration  is 
favored,  because  the  resultant  quantity  finds  in  general  no 
visible  geometrical  element  falling  in  its  line  as  the  tangent  to 
the  path  does  with  velocity.  The  projections  on  tangent  and 
normal  form  the  simplest  set  that  contains  any  segregation,  for 


The  Main  Coordinate  Systems  149 

as  we  have  once  noticed,  the  (XYZ)  set  does  not  discriminate 
but  speaks  always  of  its  own  tensors.  The  separation  on  the 
basis  that  tangential  acceleration  changes  velocity  through  its 
tensor  alone,  and  the  normal  part  changes  the  unit-vector  alone, 
is  the  most  important  early  and  familiar  instance  that  general 
ideas  of  vector  algebra  had  to  pattern  after.  The  last  of  equa- 
tions (230)  has,  as  we  are  aware,  grown  into  a  general  handling 
of  any  vector  derivative. 

113.  The  polar  components  of  acceleration  have  been  found 
to  involve  in  comparative  complexity  the  distinctive  traits  of  the 
velocity  vector  as  exhibited  by  its  derivative,  because  their 
formulation  is  guided  by  elements  foreign  to  (v)  and  borrowed 
from  the  behavior  of  the  other  vector  (r).  And  as  we  see  illus- 
trated repeatedly,  the  changes  in  any  vector  indicate  themselves 
most  directly  by  analysis  of  its  derivative  according  to  some 
leading  idea  inherent  in  the  vector  itself.  It  did  not  escape  us 
that  the  vector  (H),  for  example,  is  but  indirectly  described  by 
use  of  (w)  and  (w)  in  sections  56  and  57,  and  that  there  is  likely 
to  be  a  gain  when  the  more  direct  connection  of  (H)  and  (M)  is 
utilized. 

Before  entering  upon  any  new  considerations,  let  us  once  more 
pick  up  the  thread  at  section  89,  and  renew  the  thought  that 
(xyz)  and  (x'y'z')  can  be  read  as  projections  of  any  free  vector 
such  as  (v).  Then  equations  (154)  or  their  alternatives  made 
explicit  for  (x,  y,  z)  are  the  algebraic  statement  of  shift  for 
acceleration,  (v)  for  the  standard  frame  being  given  indifferently 
by 

V  =  ix  +  jy  +  kz;        V  =  i'x'  +  j'y'  +  k'z'.        (231) 

Also  the  details  worked  out  for  (r),  beginning  with  section  78, 
are  translatable  for  (v),  and  justify  for  instance,  as  we  can  use 
now  the  Euler  angles,  and  are  paralleling  (ro  =  0), 

3v  dy  dv  dv 


150  Fundamental  Equations  of  Dynamics 

whose  meaning  reproduced  more  briefly  in 

^  =  t(„»  +  (t  X  v)  (233) 

gives  foundation  for  our  next  useful  conclusion. 

114.  In  a  plane  curve  that  is  the  path  of  a  point  (Q),  the 
successive  orientations  of  the  tangent  can  be  said  to  arise  by  a 
continuous  turning,  whose  axis  is  the  normal  at  (Q)  to  the 
plane  of  the  path.  And  this  turning  to  which  we  assign  the 
angular  velocity  (q),  and  which  accompanies  the  progress  with 
velocity  (v)  of  (Q)  along  the  curve,  is  registered  in  its  effect  upon 
(v)  through  the  normal  acceleration  that  is  written 

v2 
^(n)  =  (o>  X  v)  =  -  Pi  -  .  (234) 

P 

The  order  of  factors  in  the  second  member  is  seen  to  direct  this 
acceleration  toward  the  local  center  of  curvature  of  the  path, 
and  the  known  geometry  introduces  the  radius  of  curvature, 
whose  standard  unit-vector  points  away  from  that  center. 
Complementary  to  this  is  the  tensor  change  in  (v)  provided  for 
by  the  tangential  acceleration  whose  natural  form  is 

dv 

V(t)  =  vi  ^-  .  (235) 

In  order  to  recast  these  statements  in  the  language  of  shift, 
let  comparison-frames  be  conceived  distributed  along  the  path 
and  with  origins  in  it,  each  in  a  permanent  configuration  with 
the  standard  frame,  its  (X')  axis  pointing  forward  along  the 
local  tangent  and  its  (Y')  axis  inward  along  the  normal,  (o)  being 
standard  as  positive.  All  such  frames  will  give  both  velocity 
and  acceleration  invariantly  with  the  standard  frame,  and  for 
each  one  as  (Q)  passes  its  origin  the  same  conditions  prevail  at 
the  epoch: 

V(x')  =  v;        V(j.')  =  V(.')  =  0.  (236) 

But  the  shift  of  origin  alone,  as  we  have  noticed  elsewhere,  being 


The  Main  Coordinate  Systems  151 

without  effect  upon  the  projections  as  vectors,  the  application  of 
equation  (233)  will  yield 

.  •/  d   .        .  dv 

^(x')  =  1  :t:  (v(x'))  =  Vi  tt  ; 

dt^  '   '  dt  '  (237) 

V(y')  =  j'(wv);         V(,')  =  0; 

consistently  with  equations  (234,  235). 

115.  But  a  space  curve  differs  from  a  plane  curve  very  much 
as  the  instantaneous  orbit  spoken  of  in  section  111  differs  from  a 
plane  orbit,  in  that  its  differential  sectors,  bounded  now  by  radii 
of  curvature  and  not  by  radius-vectors,  are  not  coordinated  into 
one  plane.  Each  is  treated  typically  like  the  uniplanar  case, 
however,  but  in  the  plane  of  its  epoch.  A  gradual  change  of 
this  plane  can  always  be  accomplished  by  an  added  turning 
about  some  axis  contained  in  each  plane  element,  the  displace- 
ments due  to  which  being  normal  to  that  element  are  merely 
superposed  on  whatever  process  is  being  completed  within  the 
plane  of  the  element  itself.  The  direction  of  each  such  axis  in 
its  individual  plane  will  be  chosen  according  to  the  particular 
condition  that  it  is  desired  to  fulfil. 

In  the  account  of  Hansen's  coordinates  it  was  proved  that  the 
designated  axis  left  both  component  velocities  (<o  x  r)  and 
(ri(dr/dt))  unaffected  by  a  rotation  about  it;  and  also  two  of 
the  three  component  accelerations.  In  the  example  before  us 
now,  it  becomes  desirable  to  leave  unchanged  the  one  velocity  (v) 
that  enters  unresolved,  and  the  entire  acceleration.  It  soon 
appears  how  this  is  attained  by  letting  each  differential  sector 
turn  about  an  axis  that  is  the  Une  of  (v)  at  the  epoch.  This  will 
add  no  new  velocity  at  any  point  Uke  (Q)  in  that  axis,  and  it 
leaves  the  acceleration  components  unaltered  because  the  supple- 
mentary term  (y'  x  v)  would  in  any  event  be  normal  to  the 
plane  element,  if  (y')  denotes  a  rate  of  rotation  about  any  axis 

in  that  plane,  and  this  term  vanishes  for  every  magnitude  of  (y') 
11 


1 52  Fundamental  Equations  of  Dynamics 

when  the  latter  is  colinear  with  (v).  Consequently  if  we  apply 
equation  (233)  again,  writing 

y  =  {i>^  +  r'),  (238) 

equations  (236,  237)  are  continued  in  validity  for  any  space 
curve,  though  derived  originally  from  uniplanar  motion.  It  is 
plain  in  what  way  the  shift  process  is  to  be  modified  when  it 
must  include  a  varying  plane  (X'Y')  for  the  osculating  circle; 
and  also  that  the  tensor  of  (y')  must  be  fitted  to  the  tortuosity 
of  the  curve,  while  (w)  is  determined  by  the  circle  of  curvature. 
The  vector  magnitude  {y')  is,  to  the  extent  shown,  external  to 
the  acceleration  problem  stated;  and  in  this  it  goes  beyond  the 
corresponding  vector  (u)  of  Hansen's  system,  as  reference  to 
equation  (228)  confirms.  The  geometry  of  space  curves,  in 
wl^ich  our  axis  (Z')  figures  as  the  binormal,  is  seen  to  build  with 
similar  ideas  to  those  just  developed. 

116.  If  a  comparison-frame  (O',  X'Y'Z')  is  moving  as  a  whole 
relative  to  the  standard  with  unaccelerated  translation  whose 
velocity  is  (vo),  and  the  velocity  of  (Q)  relative  to  (0',  X'Y'Z') 
is  (v'),  the  last  of  equations  (230)  gives  for 

dv' 

V  =  Vo  +  v',        V  =  t/v'  +  Vi'  -^ .  (239) 

And  since  by  supposition  (i'j'k')  are  here  constant  vectors,  there 
is  no  distinction  between  (t/)  relative  to  (X'Y'Z')  and  (XYZ). 
Hence  comparing  the  paths  of  (Q)  relative  to  the  two  frames,  it 
is  clear  that  the  sum  is  invariant,  if  we  add  together  each  tan- 
gential acceleration  and  its  partner  of  normal  acceleration, 
although  the  velocities  in  the  paths  are  different,  as  is  the  appor- 
tionment of  the  acceleration  between  the  two  components.  Such 
indifference  as  exists  to  the  inclusion  or  the  exclusion  of  constant 
velocities  is  often  a  helpful  fact  in  treating  of  accelerations. 
But  its  other  limitations  must  be  observed  beside  the  one  just 
indicated,  as  applying  for  example  to  power  (R-v).     If  in  this 


The  Main  Coordinate  Systems  153 

product  (R)  is  retained,  and  (v)  is  changed  to  (v'),  the  product 
is  altered  unless  (vo)  and  (R)  happen  to  be  perpendicular. 
As  the  summation 


I 


ds  =  (s  -  So)  .       (240) 


constitutes  a  rectification  of  the  path,  so  the  other  legitimate 
summation 

£(mgdt)  =  m(v-v.)  (241) 

might  be  termed  a  rectification  of  momentum.  In  each  opera- 
tion we  may  see,  by  one  waj^  of  viewing  it,  the  accumulation  of 
tensor  elements  upon  one  shifted  line  that  becomes  parallel  in 
succession  to  the  vector  elements  whose  tensors  are  thus  summed. 
But  it  does  not  explain  fully  why  the  second  summation  is 
mathematically  as  valid  as  the  first,  just  to  remark  that  each 
element  of  momentum  is  colinear  with  an  element  (ds).  The 
tensor  factors  may  be  in  any  ratio  that  varies  from  one  element 
to  another  and  distorts  the  graph.  In  addition  to  whatever 
else  can  be  said,  we  may  return  to  the  idea  of  comprehensive 
tensor  running  through  a  process  of  shift  and  observe  what 
condition  makes  an  element  of  actual  displacement  and  the 
exact  differential  of  such  a  tensor  equal,  by  obviating  that  fore- 
shortening of  each  element  and  the  telescoping  of  their  series 
that  shift  in  general  causes.  If  v/e  take  for  instance  equation 
(122)  in  connection  with  its  context,  the  condition  is  seen  to  be 
that  the  vector  product  denoted  generally  by  (y  x  V)  should  be 
perpendicular  to  the  fine  on  which  the  tensor  in  question  is  laid 
off.  This  becomes  a  specially  simplified  relation  when  the  plan 
of  shift  is  such  that  only  one  tensor  occurs.  The  polar  scheme 
contains  only  the  length  (r)  of  the  radius-vector;  the  tangent 
and  normal  resolution  only  the  tensor  of  (v),  which  may  indeed 
be  identified  with  (r)  by  the  thought  of  section  88.     In  forming 


154 


Fundamental  Equations  of  Dynamics 


the  derivative  of  (r)  or  of  (v)  under  the  form  of  equation  (137), 
(V(m))  comprises  nothing  but  the  total  derivative  of  the  tensor, 
and  the  mathematical  test  for  integrability  is  met.  If  it  were 
practically  easier  to  devise  plans  of  the  type  instanced,  without 
sacrificing  other  advantages,  there  would  be  less  hindrance  to 
forming  integrated  values  of  tensors  in  working  out  results  of 
shift. 

117.  We  shall  close  this  summary  of  our  last  system  of  point 
coordinates  by  gathering  for  record  its  most  serviceable  relations 
to  the  fundamental  quantities,  and  here  again  with  a  representa- 
tive particle  at  the  center  of  mass  of  a  body  definitely  in  mind. 
They  show  in  terms  of  projections  parallel  to  the  (X'Y'Z') 
specified  for  equations  (236,  237),  with  (xo',  yo',  zo')  added  for 
the  coordinates  in  the  standard  frame  of  the  particle  caught  in 
passage  through  the  (0')  of  the  epoch. 

Q(x')  =  Q  =  mv;         Q(,')  =  Q(,')  =  0; 

E(,')  =  E  =  imv2;         E(y')  =  E(,')  =  0; 

H  =  (xo'  +  yo'  +  ZoO  X  Q  =  +  j'(zo'Q)  -  k'(yo'Q); 


R 


')  =  ^'  (^  d^  )  '         ^(y')  =  i'(mvco);        R(,')  =  0; 

P  =  Rc.')V  =  m^v  =  ^(E(.'));  I"    (242) 

-M  =  (xo'  +  yo'  +  Zo')  X  (R(x')  +  R(y')) 

=  k'  (  mwvxo'  —  m  Tr  yo'  1 

-I-  j'  f  m  Tr  Zo'  j  -  i'(mcovzo'). 

The  expression  written  for  (M)  should  be  compared  with  the 
direct  vector  derivative  of  (H)  as  given  above  in  terms  of  the 
shifting  (X'Y'Z'). 


The  Main  Coordinate  Systems  155 

Euler's  Dynamical  Equations. 

118.  The  configuration  angles  (tjr,  ^,  ^)  have  been  associated 
with  Euler's  name  already;  and  once  more  we  follow  the  estab- 
lished custom  in  speaking  of  the  next  plan  to  be  examined  as 
Euler's,  describing  the  statements  of  it  as  his  dynamical  equations, 
and  so  contrasting  them  with  the  purely  geometrical  or  kinemati- 
cal  ideas  brought  forward  under  the  other  title.^  This  second 
group  of  Euler's  equations  constitutes  a  system  of  resolution  for  the 
dynamical  quantities  that  departs  in  one  important  respect  from 
all  the  others  that  have  preceded  it  in  the  order  that  we  are 
following.  It  has  been  constructed  with  specific  reference  to  a 
rigid  body  as  a  whole,  instead  of  being  shaped  for  one  element 
of  mass,  or  at  most  for  a  particle  at  the  center  of  mass.  The 
summation  covering  the  entire  mass  has  been  incorporated  into 
the  expressions,  as  an  integral  part  of  their  standard  form;  the 
field  of  use  for  them  is  particularly  among  those  parts  of  the  total 
quantities  that  must  fall  outside  all  plans  that  are  limited,  in 
conception  or  in  effective  and  convenient  adaptation,  to  a  par- 
ticle's translation.  Therefore  it  will  be  anticipated  that  we  shall 
deal  in  these  equations  with  that  element  of  rotation  in  the  most 
general  tj'^pe  of  motion  for  a  rigid  body,  which  is  the  obUgatory 
remainder  after  deducting  a  translation  with  its  center  of  mass. 
The  explanations  on  this  point  in  sections  48  to  63  may  be  re- 
ferred to;  also  those  in  regard  to  the  dynamical  independence 
of  the  rotation  and  the  translation,  and  the  connection  of  a  pure 
rotation  about  an  origin  with  one  about  a  moving  center  of  mass 
(see  sections  52  and  53).  Let  it  be  remarked,  in  order  to  cover 
this  aspect  of  the  situation,  that  Euler's  dynamical  equations  once 
developed  for  the  conditions  of  rotation,  are  apphcable  equally 
to  either  occurrence  of  it. 

119.  A  junction  with  previous  results  can  be  made  by  bringing 
together  the  equations  for  the  values  of  (H)  and  of  (M),  since 

» See  Note  28. 


1 56  Fundamental  Equations  of  Dynamics 

it  has  been  proved  that  moment  of  momentum  and  force-moment 
furnish  central  clews  to  guide  inquiry  among  the  phenomena  of 
rotation.  Let  the  understanding  be  that  our  analysis  attaches 
primarily  to  rotation  about  a  center  of  mass  (C),  and  that  any 
necessary  transitions  to  pure  rotation  are  to  be  adequately 
indicated. 

On  returning  to  equations  (86)  the  signs  of  mass-summation 
are  in  evidence,  and  also  of  the  general  interrelation  between 
each  component  of  (H)  and  all  three  components  of  the  rotation- 
vector  (o)),  when  an  unguided  choice  of  (XYZ)  has  been  made, 
to  which  axes  those  located  at  (C)  will  be  assumed  parallel  for  a 
beginning.  The  concept  of  (to)  as  properly  applicable  to  the 
complex  of  radius-vectors  lying  within  the  body  has  been  adopted 
profitably,  but  it  is  not  to  be  overlooked  that  a  changing  con- 
figuration of  body  and  (XYZ)  makes  the  inertia  factors  variable. 
Neither  does  parallelism  of  the  axis  of  (w)  with  one  of  (XYZ), 
permanent  or  transient,  introduce  the  lacking  symmetry  into 
these  equations.  Note,  however,  the  form  of  equation  (80), 
regard  (co)  as  parallel  to  (Z),  and  complete  the  set  of  component 
equations  thus  particularized.     They  are  for  (X'Y'Z')  at  (C), 

H(z')  =  k'(aj(z')l(z'));        H(y')  =  j'(-  co(,')/™y Vdm) ; 

(243) 
H(x')  =  i'(-  w(z')/mz'x'dm). 

Observe  the  form  of  the  last  two  components,  and  the  fact  that 
the  orienting  factors  in  them  are  coordinates. 

120.  The  commentary  of  the  last  paragraph  can  be  duplicated 
essentially  in  respect  to  equations  (89),  replacing  (H)  by  (M') 
and  (fa>)  by  (to).  Thus  if  we  next  suppose  the  axis  of  (to)  parallel 
to  (Z),  all  three  components  of  (M')  persist,  and  a  similar  differ- 
ence in  type  reappears,  between  the  first  component  and  the  two 
others.     Again  for  (X'Y'Z')  at  (C), 

M'(,')  =  k'(«(z')I(z'));        M'(y')  =  j'(-  w(z')/myVdm); 

(244) 
M'(x')  =  i'(-  aj(z')/mz'x'dm). 


The  Main  Coordinate  Systems  157 

Bringing  in  the  other  part  (M")  of  the  total  force-moment  does 
not  better  the  symmetry,  neither  of  the  last  equations  nor  of 
their  parent  equations,  since  in  reliance  upon  equations  (75,  76) 
we  find 

M"  =  (a>  X  H).  (245) 

These  observations  multiply  reasons  for  appropriating  the 
principal  axes  at  (C)  in  a  selective  choice  of  (X'Y'Z')  for  any 
one  epoch,  and  then  perpetuating  whatever  advantages  are 
reaped,  by  introducing  a  shift  that  is  so  regulated  that  the  same 
three  hues  of  the  body  which  are  its  principal  axes  for  (C)  shall 
always  be  taken  to  mark  or  indicate  the  configuration  of  the 
fixed  frame,  in  terms  of  whose  projections  or  components  of  the 
quantities  in  question  the  equations  are  to  be  written.  The 
case  for  these  principal  axes  is  strengthened  when  equation  (88) 
adds  kinetic  energy  to  the  expressions  in  this  way  simplified; 
and  when  we  reflect  that  within  the  scheme  now  proposed,  the 
inertia  factors  are  reduced  from  six  in  number  to  three  that  are 
the  principal  moments  of  inertia,  and  that  the  triplet  retains  the 
same  values  as  the  axes  under  this  scheme  shift.  The  general 
case  is  to  be  supposed,  where  there  are  no  more  than  three 
principal  axes  at  (C),  and  the  momental  ellipsoid  is  not  one  of 
rotation. 

In  view  of  the  role  about  to  be  assigned  to  them,  a  specialized 
notation  referring  to  principal  axes  is  called  for,  and  we  shall 
meet  that  need  first  by  using  (A,  B,  C)  to  denote  both  the 
magnitudes  of  the  principal  moments  of  inertia  and  the  axes 
with  which  they  are  associated.  As  magnitudes,  (A,  B,  C)  are 
scalar  factors  in  equations.  They  are  associated  with  hues  and 
not  with  either  one  direction  in  those  Unes,  so  they  are  not 
vector  tensors.  As  axes  for  specifying  configuration,  (ABC) 
designate  by  convention  one  direction  in  each  line.  The  cycle 
order  is  as  they  stand  written,  so  that  in  the  zero  of  configuration, 
(A)  is  parallel  to  (X),  (B)  to  (Y),  and  (C)  to  (Z).     The  axis  of 


158  Fundamental  Equations  of  Dynamics 

(C)  is  then  (Z')  of  our  preceding  notation,  and  it  has  at  any 
epoch  the  angular  coordinates  (ijr,  •&).  The  third  angular  dis- 
placement (^)  is  about  the  (C)  axis  itself.  (See  section  93.) 
Secondly,  projections  of  any  vector  upon  the  principal  axes  will 
be  denoted  as  illustrated  for  (w)  and  (&>)  thus: 

W   =   W(a)   +  W(b)   +  W(c);  0>   =   fa>(a)   +  <I>(b)    +  "(c);         (246) 

and  the  corresponding  unit-vectors  by  (ai,  bi,  Ci). 

Utilizing  this  notation,  the  equations  brought  under  review 
above  are  reduced  to  the  forms 

H  =  o)(a)A  +  (O(b)B  +  fa>(c)C; 

(247) 
M'  =  w(a)A  +  «(b)B  -}-  <b(c)C; 

M"  =  ai(a>(b)CO(c)C  —  C0(c)C0(b)B) 

+  bi(cO(c)CO(a)A   —    aj(a)aj(c)C) 

+  Ci(w(a)CO(b)B  —  co(b)a)(a)A);    (248) 

E    =   MAco2(a)    +  Bco2(b)    +  Ca,2(e)].  (249) 

And  this  yields  for  the  similar  components  of  the  total  moment 

(M) 

M(a)  =  ai[a>(a)A  +  W(b)a)(c)(C  —  B)];  1 

M(b)  =  bi[ci(b)B  +  co(e)co(a)(A  -  C)];  y  (250) 

M(c)   =  Ci[a)(c)C  +  co(a)a)(b)(B  —  A)].  J 

The  sequence  of  ideas  by  which  these  specialized  equations  have 
been  reached  should  be  attentively  scrutinized,  also  the  inter- 
pretation of  the  combinations  at  this  stage.  Equations  (250) 
are  evidently  valid  at  any  one  epoch,  and  can  be  evaluated  if 
these  elements  are  known  at  that  epoch: 

(1)  The  orientation  of  the  axes  (ABC)  in  (XYZ),  and  the 

magnitudes  (A,  B,  C); 

(2)  The  vector  (6)  in  tensor  and  orientation; 

(3)  The  vector  (w)  in  tensor  and  orientation. 


The  Main  Coordinate  Systems  i   159 

121.  In  order  to  supply  some  other  profitable  details,  and  to 
put  another  link  in  the  connections  of  these  equations  with 
general  forms,  we  shall  recur  to  equations  (86)  and  differentiate 
with  regard  to  time,  the  first  of  them  for  a  sample.  It  is  funda- 
mental that  the  result  must  represent  the  projection  of  (M) 
upon  (X),  the  latter  being  taken  arbitrarily;  and  that  with  base- 
point  at  (C)  all  moments  must  be  reducible  to  couples,  all  net 
force  being  absorbed  into  the  translation.  (See  section  51.)  The 
conspicuous  complication  in  this  derivative  is  a  lesson  about 
what  principal  axes  avoid,  for  we  find 


H(x)  =  M(^)  =  i|  I(x)^(w(^))  -f  co(x)^(I(j)) 

.    dx     ,  r     dy  , 

-  '«'(y)/m^  ydm  -  c<j(y)/„,x^-dm 

-  ^  (aj(y))/„>xydm  -  aj(z)/m  ^  zdm 

-  w(z)/mX  ^dm  —  t:  (co(,))/mZxdm  \  .     (251) 

In  the  third  member,  the  third,  fourth,  sixth  and  seventh  terms 
are  to  be  further  expanded  by  use  of  the  velocity  relations  for 
rotation, 


dx  dy 

^  =  (^(y)2  -  w(z)y;  ^  =  w(z)X  -  w(x)z; 

dz 

^  =  co(i)y  -  co(j-)X. 


(252) 


When  the  axes  (XYZ)  are  particularly  chosen  to  be  the  set  (ABC) 
in  its  position  at  the  epoch,  all  terms  can  be  struck  out  that 
contain  as  factors  the  integrals  known  as  products  of  inertia. 
And  this  choice  cancels  the  second  term  in  the  third  member 
also.  Because  for  all  sets  of  orthogonal  axes  at  the  same  origin 
we  have 


160  Fundamental  Equations  of  Dynamics 

I(x)  +  I(y)  +  I(z)  =  2/inrMm  (an  invariant  magnitude);    (253) 
and  hence  during  relative  displacement  of  body  and  (XYZ), 

ft  ^^^^^^  +  ft  ^'^'^'^  +  ^t  ^'^'-'^  ^  ^-  ^^^^^ 

But  for  the  longest  and  for  the  shortest  axis  of  the  momental 
ellipsoid,  corresponding  to  the  least  and  the  greatest  principal 
moment  of  inertia,  the  condition  of  maximum  or  minimum  re- 
moves two  terms  separately  from  the  above  equation  of  condi- 
tion, which  then  proves  that  a  stationary  value  of  moment  of 
inertia  enters  for  the  third  principal  axis  also. 

After  removing  all  the  terms  of  indicated  zero  value,  there 
remains 

H(x)  =  ai  I  I(x)  ^  (co(x))  +  aj(y)a>(z)/m(y2  +  x2)dm 

-  co(z)co(y)/m(z2  +  x2)dm  \  ,     (255) 

for  comparison  with  the  first  of  equations  (250).  The  two  state- 
ments harmonize  completely,  if  we  insist  upon  the  identity  of 
meaning  for  the  expressions 


4dt^''^^>)J' 


w(a),        i     ^  (co(x))     ;        [(A)  and  (X)  parallel.] 

they  are  both  representative  of  the  projection  of  the  vector  (w) 
upon  (A)  or  (X).  The  comparison  for  the  two  other  pairs  of 
equations  is  to  be  made  similarly. 

122.  The  next  step  in  progress  releases  equations  (250)  from 
this  one  reading  of  their  symbolism,  and  lays  a  foundation  for 
the  equivalences 

"(a)  =  ai  T7  (t«J(a)) ;        b>(b)  =  biT7(aj(b)); 

"^^  ^  "^^  (256) 

"(c)     ^    ^1^  (W(c))> 

where  the  second  members  are  to  be  recognized  as  components 


The  Main  Coordinate  Systems  161 

of  (V(m))  in  equation  (137),  for  application  to  the  derivative  (<I>) 
as  expressed  under  a  process  whose  shift  rate  is  marked  by  the 
axes  (ABC).  Since  these  are  definite  Hnes  of  the  body,  they 
must  conform  to  its  rotation-vector  («),  and  we  have  in  this 
shift  another  example  of  cancelled  correction,  for 

W    =   <J(m)   +    (o>  X  (o)    =    ^1  jl   ('•'(a))    +  bl   jT    ('•'(b)) 

+  Cl^(a,(e)),       (257) 

where  the  tensors  in  the  third  member  have  taken  on  a  new 
shade  of  interpretation.  They  have  become  the  generalized 
values  for  the  shifting  axes,  instead  of  being  particularized  single 
values. 

But  there  is  one  more  consequence  in  this  direction  that  still 
remains  to  be  formulated,  and  that  can  be  drawn  from  the 
expression  in  equations  (247)  for  moment  of  momentum  which 
can  now  be  conceived  as  continuously  valid  and  differentiated, 
due  allowance  being  included  for  the  changing  orientation  of  the 
projections  that  make  up  the  total.     We  can  write 

EiA  ^  (co(a))  +  biB  ^  (w(b))  +  CiC  ^  (aj(c)) 

+  (<o  X  H),     (258) 

whose  separation  into  components  restates  equations  (250),  after 
incorporating  into  the  latter  the  transitions  of  equations  (256). 
The  forms  derived  by  either  line  of  procedure  are  Euler's  dynami- 
cal equations,  whose  establishment  with  the  means  at  their 
inventor's  disposal  must  always  be  rated  as  a  remarkable  achiev-e- 
ment.  It  is  in  addition  moreover  remarkable  that  the  segre- 
gation according  to  the  terms  of  equation  (258),  which  is  more 
nearly  mathematical  in  its  origin,  is  also  a  separation  that  splits 
the  force-moment  into  parts  with  a  plain  and  important  difference 


162  Fundamental  Equations  of  Dynamics 

of  physical  effect;  and  the  beginning  made  in  section  55  was 
with  design  selected  in  order  to  dwell  upon  that  fortunate  chance. 
A  conclusive  proof  of  the  equations  in  very  few  lines  can  evi- 
dently be  extracted  from  the  material  that  has  been  discussed 
here  with  greater  expansion;  but  a  demonstration  may  become 
too  brief  to  be  effective  for  insight,  in  a  matter  that  has  wide 
general  bearings,  so  the  detail  is  probably  not  superfluous. 

123.  Among  the  uses  of  Euler's  equations,  the  predominant 
type  of  rigid  body  whose  rotation  is  to  be  investigated  is  likely 
to  show  a  certain  symmetry,  whose  representation  in  the  mo- 
mental  ellipsoid  gives  equality  to  two  axes  of  the  latter.  This 
must  convert  the  general  ellipsoid  into  one  of  rotation  with  a 
symmetry  axis;  the  known  consequence  being  that  all  per- 
pendiculars to  that  symmetry  axis  at  the  center  of  the  rotational 
ellipsoid  become  principal  axes  with  equal  moments  of  inertia. 
This  combination  arises  if  the  rotating  body  itself,  being  homo- 
geneous in  material,  has  an  axis  of  symmetry;  and  bodies 
designed  for  rapid  revolution  are  usually  turned  in  a  lathe.  But 
it  is  clear  that  a  prism  of  square  cross-section,  as  well  as  a  circular 
cylinder,  would  manifest  its  symmetry  in  a  momental  ellipsoid 
of  rotation.  And  Euler's  dynamical  equations,  being  concerned 
with  distribution  of  mass  only  as  recorded  in  principal  moments 
of  inertia,  would  not  discriminate  between  the  two  cases,  granted 
the  magnitudes  (A,  B,  C)  are  severally  equal  in  them. 

It  is  proposed  next  to  reconsider  equations  (250)  in  the  light 
of  this  possibility,  designating  (C)  as  the  axis  of  symmetry  of 
the  momental  ellipsoid  for  (C),  with  the  corollary  that  the 
magnitudes  (A)  and  (B)  are  equal;  their  common  value  we  can 
call  (A).  If  now  the  axes  of  (A)  and  (B)  are  stiU  definitely 
located  as  lines  of  the  body,  whose  rotation-vector  (y)  is  identical 
therefore  with  (to)  for  the  body,  no  essential  change  appears  in  the 
equations  except  dropping  out  the  last  term  of  the  third.  Espe- 
cially equations  (256)  that  are  determined  by  the  equality  of 


The  Main  Coordinate  Systems  163 

(y)  and  (a>)  are  available  as  before.  However  all  lines  of  exposi- 
tion in  reaching  Euler's  equations  must  set  the  adoption  of 
principal  axes  in  the  central  place,  and  not  the  equality  of  the 
rotation-vectors.  So  by  multiplying  the  number  of  principal 
axes  the  condition  of  symmetry  enables  choice  to  be  variouslj'' 
exercised  and  yet  range  among  them,  though  the  auxiliary  equal- 
ity be  abandoned  and  a  relative  motion  through  the  material 
of  the  rotating  body  be  permitted  to  the  principal  axes  that  have 
been  selected.  It  is  clear  that  the  assumed  relations  limit  the 
difiference  between  (y)  and  (o)  to  a  turning  about  (ci)  that  is 
also  (^i);  but  to  this  element  it  remains  free  to  assign  any 
magnitude.     The  expression  of  that  freedom  is 


d^  d??  d<p 

d^  dt?  /,   d^\ 


(259) 


where  (k)  may  have  any  positive  or  negative  value.  Euler's 
equations  proper  given  for  (k  =  1)  have  been  put  before  us 
already;  and  we  shall  add  for  consideration,  among  the  gener- 
alized Euler  forms  suggested  by  the  last  equation,  only  that 
modification  which  becomes  necessary  when  the  value  of  (k)  is 
taken  at  zero.  This  supposition  happens  to  offer  some  special 
advantage  in  handling  combinations  like  a  gyroscope  under 
control  by  weight  moment,  and  the  earth  as  affected  by  a  gravita- 
tion couple  due  to  its  spheroidal  figure. 

124.  Let  us  mark  the  change  of  plan  by  using  (A',  B')  to 
denote  those  principal  axes  that  are  now  substituted  for  (A,  B), 
recollecting  first  that  as  moments  of  inertia  all  four  magnitudes 
are  equal,  and  secondly,  that  (C)  is  common  to  both  sets  of 
axes.  Then  as  a  reminder  of  the  needed  revision  in  equations 
(256)  we  can  write 


164 


Fundamental  Equations  of  Dynamics 


W    =    ai'    37   (W(a'))    +   bi     TT   (W(b'))    +    Ci  -77    (W(c)) 


dt 


dt 


'^dt 


+ 


((^^^  +  ^^¥)x")-    (2^«) 


In  equations  (193)  are  recorded  values  for  those  components 
of  («)  which  accord  in  directions  with  the  present  specifications 
for  (A'B'C);  and  in  equations  (191)  of  section  101  the  line  of 
development  caused  us  to  put  down  in  terms  of  (tjr,  -d,  ^)  the 
first  three  entries  on  the  right  hand  of  equation  (260).  It  seems 
advisable  to  clinch  the  comparison  in  respect  to  equations  (256, 
257)  by  developing  here  for  that  resolution  the  general  com- 
ponents of  (w),  and  lastly  confirming  the  harmony  of  the  two 
sets  at  their  coincidence  that  occurs  for  (^  =  0).  These  are  the 
first  details: 


w(a)  =  ^1  ^  ((^w)  =  a 


.[ 


d^ 
dt2 


d^  d<p 
dt  dt 


dV    .  .  di/'dt? 

+  37^  sm  I?  sin  ^  -f-  —  37  cos  ^  sin  cp 


dt2 


dt  dt 


.   ^^  dip  "I 

+  dtdi'^°'^'°'^J' 


■■[ 


d^d 


dt?  d<p 


«b(b)  =bi^(a,(b))  =bi[^-^sm^--^cos^ 
dV    .  di^dt? 

+  -r—  sin  t?  COS  (f  -\-  -TT  37"  cos  t?  COS  <p 

at''  at  dt 


diAd 
dt  dt 


<P    .         •        1 
-  sin  t?  sin  ^     ; 


«(c)  =  <^i  ^  ("(c))   = 


fdV 
-''adt^ 


dV 
+  TIF  cos  t? 


dt2 


d^(M 
dt  dt 


sin  t?     ; 


(261) 


The  Main  Coordinate  Systems  165 

the  values  to  be  differentiated  in  the  second  members  being  duly- 
identified  in  the  survey  that  equation  (181)  has  put  together. 
What  remains  of  the  results  last  written  when  they  are  particu- 
larized for  the  condition  (§  =  0);  with  (<«)(&')),  (o>(b'))  obtained 
by  a  corresponding  resolution  of  (<o),  fills  out  the  more  general 
form  of  Euler's  equations, 

M(a')  =  a/[w(a')A'  +  co(b')co(c)(C  -  B')]; ' 

M(b')  =  b/[ci(b')B'  +  a;(c)co(a')(A'  -  C)];  •  (262) 

M(c)  =  Ci[aj(c)C]; 

the  necessitated  companion  being  the  equalities  of  magnitude 

A  =  A'  =  B  =  B'.  (263) 

Finally  the  components  of  (w)  that  match  the  above  statement 

being  added: 

,fd^\  ^  ,/#    .        \ 

w(a')  =  ai  I  ^  )  ;        «(b')  =  bi  I  ^  sm  z?  I  ; 

(264) 


"(c) 


fd<p       dxl^  \ 


such  advantage  as  this  alternative  formulation  possesses  on  the 
kinematical  side  is  made  to  appear.  Dynamically  something  is 
contributed  to  a  preference  for  it  when  the  resultant  force- 
moment  is  a  vector  that  lies  continually  in  the  line  of  the  axis 
(A').  A  preliminary  examination  of  the  instances  quoted  above 
shows  that  they  lend  themselves  unconstrainedly  to  this  analysis 
which  will  be  found  applied  in  section  127. 

125.  On  the  surface  the  constant  reference  to  ((^)  and  (&>), 
either  in  their  totals  or  through  differently  designated  sets  of 
their  components,  is  apt  to  leave  a  misleading  impression  that 
they  are  pivotal  quantities  in  any  investigation  where  Euler's 
equations  are  employed.  It  seems  worth  while,  therefore,  to 
put  in  stronger  light  the  primary  emphasis  of  equation  (258) 


166  Fundamental  Equations  of  Dynamics 

upon  changes  that  are  going  on  in  the  moment  of  momentum 
vector  (H).  The  separation  in  the  second  member  there  fits  a 
line  of  demarcation  between  changes  in  magnitude  and  in  direc- 
tion, since  the  first  group  of  terms  is  by  the  connections  that  have 
been  estabhshed  for  it  a  magnitude  derivative  of 

H  =  ai(Aco(a))  +  bi(Aw(b))  +  Ci(Caj(c)),  (265) 

though  distorted  from  its  value  as  reckoned  in  the  standard 
frame  by  shift  of  the  axes  (ABC).  But  just  that  shift  is  in- 
dispensable, as  we  have  insisted,  in  order  that  the  properties  of 
principal  axes  may  prune  the  cumbrous  algebraic  expansions 
into  maximum  brevity.  Where  a  corrected  segregation  for  (H) 
into  changes  of  magnitude  and  of  direction  entails  a  sacrifice  of 
the  gain  by  using  (ABC),  the  balance  of  choice  leans  always  one 
way;  that  much  of  dynamical  indirectness  in  Euler's  equations 
is  condoned.  But  there  is  an  increasing  tendency  and  a  whole- 
some one,  to  put  their  dynamical  sense  to  the  front,  letting  (o>) 
and  (6)  fall  into  a  subordinate  importance,  derived  in  large 
degree  from  the  clews  they  furnish  to  (M)  and  to  the  course  of 
events  for  (H).  It  was  less  easy  to  do  this  under  the  older  forms 
of  Euler's  day,  but  it  is  facilitated,  as  has  perhaps  been  made 
convincingly  apparent,  by  a  vector  algebra  that  follows  so 
intimately  the  history  of  vector  quantities. 

126.  Naturally  the  thought  has  suggested  itself  to  inquire  after 
a  scheme  modeled  upon  the  resolution  of  force  into  a  tangential 
and  a  normal  component,  for  application  to  moment  of  momen- 
tum. One  main  obstacle  is  not  difl5cult  to  detect,  for  after  indi- 
cating the  start  in  parallel  to  the  other  procedure, 

dH\ 
dt  ) 


H  =  hi(H) ;         H  =  hi(H)  +  h:  (J  ^  )  ;  (266) 


it  is  noticeable  first  that  (H)  cannot  be  assumed  to  fall  in  a 
principal  axis,  and  secondly  that  no  data  for  (hi)  are  available 


The  Main  Coordinate  Systems  167 

from  geometrical  sources.  Therefore  the  longer  forms,  for  (H) 
in  equation  (86)  and  for  (dH/dt)  in  equation  (251)  must  be  used, 
and  the  expressions  must  be  encumbered  with  an  added  angular 
velocity  for  (hi).  Introduction  of  (XYZ)  gives  no  help,  nor  of 
the  partial  time-derivatives  that  rely  upon  holding  (ABC)  sta- 
tionar3^  Either  leaves  commingled  the  parts  that  are  sought 
distinct. 

But  one  resolution  of  force-moment  can  be  carried  through 
that  is  different  from  Euler's  and  yet  has  aspects  that  recom- 
mlend  it.  This  is  contrived  so  that  one  component  is  taken  in 
the  axis  of  ((o)  at  each  epoch,  and  arranged  otherwise  as  will  be 
explained  presently;  approaching  in  plan  the  tangential  resolu- 
tion of  force  in  so  far  as  (w)  and  (v)  can  be  said  to  bear  similar 
relations  to  the  two  aims.  It  has  the  merit  besides  of  piecing 
out  the  usual  discussion  of  rotation  about  a  fixed  axis,  by  giving 
recognition  to  those  supplementary  terms  which  disappear  on 
fixing  the  axis  about  which  the  body  is  rotating. 

Return  to  the  value  of  (M")  in  equation  (75)  and  of  (M') 
formed  by  mass-summation  of  equation  (82),  and  assemble  their 
respective  contributions.  Let  (u)  denote  the  rate  of  change  in 
direction  of  (o>),  so  that  with  unit-vector  (<oi)  we  have 


"  =  "^(^) 


^j  +  (uxco);  (267) 

where  (u)  must  be  perpendicular  to  (w);  and  subdivide  (r)  as 
shown  by 

r  =  r(„)  +  r'.  (268) 

Then 


d>-r 


Tdco  1 

M(„)  =  6)1    Tr/m(r-r)dm  —  /mr(„)(w-r)dm     , 

=  (  wi  ^  +  (u  X  <o)  J  •  (ru)  -t- r') 


(269) 


^l^ru))  +  (ux(o).r'.     (270) 


12 


168  Fundamental  Equations  of  Dynamics 

Identify  (Z')  with  (w),  and  (u)  witk  (Y')  in  direction,  giving 

rdo)  2  ~l 

j7  fm(r^  -  z'  )dm  —  uw/mz'x'dm 

fdoj  "1 

=  0)1     jT  I(^'J  -  uoj/xnzVdm     .  (271a) 

In  the  plane  (X'Y')  we  have  to  consider 

-  /m(o)  X  r)(fa>-r)dm  -  /„!'(« -r) dm  +  (u  x  to)/m(r-r)dm,    (272) 
from  which  are  gathered  without  difficulty 


M 
M 


(y')  =J'[- 


coVmy'z'dm  -  ^  /mz'x'dm  +  uwl(x') 


]. 


coVmz'x'dm  —  -TT  /my'z'dm 


—   Uw/mX 


ydm]. 


(271b) 


Noteworthy  is  the  extent  to  which  equations  (271)  are  reduced 
by  symmetries,  though  (u)  is  not  zero,  as  well  as  the  reappearance 
of  the  elementary  form  when  (u)  vanishes.  Dissection  of  these 
moments  shows  almost  immediately  the  force  elements  at  (dm) 
in  components  parallel  to  our  (X'Y'Z')  to  be 


dR(.')  =  i'[-^y'-  ^vjdm; 


Tdw 
dR(y')  =  j'     ^  x'  -  uwz'  -  a;2y 


'J  dm; 


(273) 


dR(z')  =  fc)i(ucoy')dm; 

which  should  be  connected  also  with  equation  (72)  by  direct 
projection  upon  (X'Y'Z'),  and  by  applying  the  proper  shift 
process  to  (H),  determined  by  the  elements  (to,  u)  on  the  same 
line  as  sections  111  and  115  develop. 


The  Main  Coordinate  Systems  169 

Regular  Precession  and  Rotational  Stability. 

127.  The  aim  and  scope  of  these  discussions  could  not  attempt 
to  include  many  particular  requirements  of  individual  problems 
without  transgressing  the  boundary  set  by  their  intention,  which 
is  guided  rather  toward  preparation  for  more  generic  or  recurrent 
needs.  It  is,  therefore,  only  because  the  dynamical  features  of 
gyroscopic  action  are  generally  acknowledged  to  be  typical 
within  a  comparatively  broad  range,  that  some  space  is  con- 
ceded to  examination  of  them.  But  though  this  carries  us 
beyond  the  stage  of  laying  out  a  plan  and  somewhat  into  exe- 
cution of  it,  it  is  proposed  not  to  go  far  in  that  direction,  nor  to 
speak  of  more  than  two  topics  that  are  critical  points  in  the 
general  perspective.  The  first  of  these  takes  the  form  of  a 
deliberate  inquiry  into  the  circumstances  of  that  adjustment  to 
steadied  motion  which  is  described  with  a  phrase  of  wide  ac- 
ceptance as  regular  precession,  and  about  which  as  a  center  so 
much  else  can  be  made  to  figure  as  a  disturbance  of  it  or  a  de- 
parture from  it.  And  the  second  is  devoted  to  laying  bare  the 
play  of  dynamical  factors  that  operates  to  produce  rotational 
stability.^ 

The  arrangement  of  the  gyroscope  is  assumed  to  give  it  a  pure 
rotation  about  a  fixed  point  (0),  that  is  now  taken  as  origin  for 
axes  like  (A'B'C),  the  last  named  being  an  axis  of  symmetry, 
the  shift  rate  for  the  set  being  as  agreed  in  section  124,  and  the 
zero  of  configuration  being  marked  by  coincidence  of  (A'B'C) 
with  (XYZ),  where  the  (Z)  axis  is  chosen  vertical  and  down- 
wards. The  total  controlling  force-moment  is  supposed  to  be 
furnished  by  weight,  the  standard  frame  being  fixed  relatively 
to  the  earth,  and  the  gyroscope  has  universal  joint  freedom  at  (O). 
For  its  rotation-vector  (o))  then,  the  two  equivalents  have  been 
supplied, 

•  See  Note  29. 


170 


Fundamental  Equations  of  Dynamics 
d(A  dt^  dip  ,/dt?\ 


+ 


b/(^sin»)  +  c.(^  +  fcos»).    (274) 


For  regular  precession  the  conditions  that  obtain  are 


d«?      ^ 


d\f/    d(p 

~rr  )  TT  )  «^j  constant: 

dt     dt 


or 


(275) 


w(a')  =  0;         w(b'),  w(o),  constant. 
And  in  order  to  standardize  values,  attach  the  further  conditions 


d<p 
df>«' 


0  <t?  <2; 


A'  >  C. 


(276) 


Then  the  weight  moment  is  negatively  directed  in  the  axis  (A'), 
and  with  understandable  notation  the  application  of  equations 
(262)  to  this  adjustment  shows  the  following  scheme  of  specialized 
values : 


(277) 


r     d\l/  d<p 
ai'(-  Wr  sin  t?)  =  a/    A^-^  dt  ^^^  ^ 

0  =  bi'  [zero]; 
0  =  Ci  [zero]. 

It  is  a  clear  matter  of  algebra  that  the  first  equation  is  satisfied  for 
sin  i}  =  0; 


or  for 


d^P_  _  co(c)C  ±  V(Ca;(c))2  +  4WrA  cos  t? 
dt  ~ 


2A  cos  I? 
or  in  another  expression  of  it  for 


d^ 
dt 


C^±-J(c^^y+4W?(A-C)cos^ 

2(A  -  C)  cos  "^ 


(278) 


The  Main  Coordinate  Systems  171 

Putting  aside  for  the  moment  the  first  root,  our  questioning 
begins  with  ascertaining  the  dynamical  double  process  that 
finds  expression  in  the  two  signs  of  the  second  root  and  that 
shows  to  inspection  in  either  form  under  the  assumed  relations 
of  value,  a  quicker  rotation  about  (Z)  and  a  slower  rotation  of 
opposite  sign  as  possible  adjustments. 

128.  It  lies  on  the  surface  that  while  regular  precession  con- 
tinues the  vector  (H)  can  be  changing  its  orientation  only  and 
not  its  tensor,  and  that  since  (H)  must  always  be  contained  in 
the  plane  (B',  C),  the  applied  force-moment  must  in  the  adjust- 
ment meet  the  condition 

a/(-  W?  sin  t?)  =  ^ijri^  X  h)  (279) 

equally  at  the  quicker  rate  and  at  the  slower  rate  of  rotation 
about  the  vertical  axis.  For  the  explanation  how  this  can 
occur,  we  shall  look  upon  the  moment  of  momentum  as  built  up 
by  superposition,  following  the  second  member  of  equation  (274) 
in  its  elements  which  are  now  the  first  and  third  only.  The 
contribution  from  the  principal  axis  (C)  and  its  horizontal  part 
effective  here  in  (M)  let  us  write 


H,.>=c.(c^^);        N'  =  n.(c^^in^). 


(280) 


Then  having  excluded  (Z)  from  being  a  principal  axis  by  the 
suppositions  laid  down  in  the  inequalities  (276),  the  second  instal- 
ment of  (H)  must  allow  for  both  a  vertical  and  a  horizontal  part,, 
the  latter  being  contained  in  the  plane  (Z,  C) ;  and  it  alone  is  ef- 
fective in  (M);  call  it  (N").  The  total  effective  component  of 
(H)  for  the  vector  product  of  equation  (279)  is  accordingly  an 
algebraic  sum 

N'  H-  N"  =  Hi  r±  C ^  sin  t?  ±  (A  -  C)  ^  sin  t?  cos  t?  ,  (281) 
the  part  (N")  being  readily  evaluated  to  confirm  this. 


172  Fundamental  Equations  of  Dynamics 

129.  It  is  hext  apparent  from  the  cycle  order  that  the  rotation 
about  (Z)  must  be  negative  in  order  that  both  terms  within  the 
parenthesis  may  first  point  the  same  way  relatively  to  (ni)  for 
our  fixed  assumptions,  and  secondly,  give  by  the  vector  product 
that  negative  orientation  in  (A')  which  the  operative  and  nega- 
tive weight-moment  demands.  So  the  standardized  form  in  the 
circumstances  becomes 

M(a')  =  (^tlri-xHJ 

=  airC^-^sin^  -  (A  -  C)  f  ^  Y  sin  ??  cos  ^1 .     (282) 

It  is  patent  how  elastic  the  constancy  of  this  algebraic  sum  can 
be  made,  or  of  its  equivalent  vector  product;  large  (N'  +  N") 
and  slow  rotation,  or  smaller  (N'  +  N")  and  quicker  rotation. 
With  equation  (281)  besides  to  show  reversal  of  the  rotation  about 
(Z)  converting  a  numerical  sum  into  an  algebraic  one,  all  other 
elements  being  held  unchanged.  But  leaving  those  details  as 
covered  sufficiently,  it  behooves  us  to  note  in  equations  (278) 
that  each  double  value  has  its  own  common  quantities  that  are 
not  entirely  reconcilable.     Since 

/d^      diA  \ 

"(e.)  =  ^i(dt  +dt  ^°'^j'  ^^^^^ 

the  first  member,  together  with  both  (??)  and  (d^/dt),  cannot  all 
remain  unchanged  while  the  rotation  about  (Z)  is  made  fast  or 
slow.  Equation  (281)  has  tacitly  taken  one  choice;  but  (<i>(c)) 
is  a  standard-frame  quantity,  whose  constancy  in  magnitude 
moreover  is  assured  under  the  third  of  equations  (262)  whenever 
(M(c))  is  zero.  We  might  then  attach  our  thought  preferably  to 
the  first  form  in  equations  (278),  and  recast  the  result  thus: 


M 


(a') 


=  air('^sina)(c<o,.,  -  A^cos*]];     (284) 


The  Main  Coordinate  Systems 


173 


in  which  the  possibihties  of  varying  factors  in  a  constant  product 
reappear,  with  (i?)  and  (w(c))  barred  from  change.  It  will  be 
noticed  finally  that  either  more  direct  derivation  of  result  corre- 
sponds exactly  with  the  terms  to  which  the  first  of  equations 
(277)  reduces,  so  our  analysis  reversed  could  be  applied  im- 
mediately to  the  latter.  It  ought  to  be  said  about  the  realization 
of  conditions,  that  the  spin  round  the  (C)  axis  is  usually  pre- 
ponderant heavily  in  magnitude,  and  for  this  reason  the  observed 
rotation  about  the  vertical  with  a  negative  weight  moment  is 
normally  retrograde,  the  necessary  high  rate  for  the  contrary 
rotation  being  practically  unattained. 

130.  Let  equations  (262)  next  be  released  from  their  restriction 
to  that  adjustment  whose  relations  are  now  ascertained.  Then 
with  repetition  of  the  idea  put  forward  in  the  connection  of 
section  56  there  can  be  a  rearrangement  in  this  instance,  too, 
that  will  describe  the  general  action  in  terms  of  a  deviation  from 
adjustment  as  a  convenient  basis  for  exhibiting  the  consequences 
in  the  light  of  a  disturbance.  Re-establishing  their  unspecialized 
character,  equations  (277)  will  be  written 


ai'(—  Wr  sin  ??)  =  a 


•■[ 


,  d2?>      dip  .       ^ 
^dF  +  dF^^"^^"^"^ 


=  ^i4. 


-T-  COS  ^A  -TT  sm 
dt  dt 


'I 


(w(b')) 


0 =cir 


diA  ,  dt} 

+  dt^°^^^dt 

C^^(cO(e)) 


-d^CcO(e)J, 


d^  ,  dxf/   .  di/'   .        ,  di}' 

+  dt"^d^^^^^-dt^^^''^dt 


(285) 


But  all  the  items  there  put  down  only  elaborate  still  the  one 


174  Fundamental  Equations  of  Dynamics 

dynamical  fact  that  no  vector  change  in  moment  of  momentum 
is  ever  being  produced  except  the  increment  along  the  instan- 
taneous position  of  the  (A')  axis,  which  is  that  of  (#i).  Denote 
the  projection  of  (H)  upon  the  plane  (Z,  C)  by  (H'),  and  the 
first  of  the  three  expressions  can  be  put  in  these  equivalent  forms : 

aA-W?sm.)=a,'[f^(A^)]  +  (*.^xH'); 


(286) 


The  first  statement  is  read  that  the  weight  moment  devotes  to 
changing  magnitude  for  the  component  of  (H)  in  its  own  line 
whatever  margin  remains  after  providing  for  continuance  of 
change  in  direction  for  the  rest  of  (H).  And  the  second,  that 
the  deviation  of  the  actual  moment  (M)  from  the  adjustment 
moment  (M(o))  required  for  prevailing  values  is  registered  in  a 
process  of  change  for  (d).  The  indicated  preemption  claim  of 
the  changes  in  direction  has  a  certain  figurative  shading,  we  may 
allow,  but  a  certain  truth  also;  because  those  affect  quantities 
at  their  existent  values  for  the  epoch,  whereas  the  quantities 
that  are  changes  in  magnitude  are  called  into  being  and  not 
present  already.  And  so  with  the  second  form  of  statement: 
the  section  referred  to  concedes  that  the  subtracted  force-moment 
in  the  first  member  may  be  declared  nominal  or  mathematical; 
but  both  points  of  view  above  are  dynamically  suggestive  and 
to  be  entertained  as  a  mental  habit. 

The  other  equations  of  the  group  (285)  set  forth  the  kinematical 
complications  that  ensue  because  nothing  dynamical  is  effective 
in  those  lines.  They  give  foundation  for  important  and  inter- 
esting studies  that  are,  however,  only  to  be  alluded  to  here; 
we  shall  content  ourselves  with  insisting  once  more  upon  the 
thought  of  sections  56  and  57.  At  the  regular  precession  adjust- 
ment every  term  in  the  second  members  of  these  equations 


The  Main  Coordinate  Systems  175 

vanishes  separately  and  they  become  a  blank  recording  nothing. 
Now  they  sum  up  algebraically  to  zero,  though  the  individual 
terms  need  not  vanish;  but  they  are,  in  a  sense  to  be  understood 
with  due  Hmitations,  as  empty  of  phj^sical  content  as  ever;  they 
chronicle  only  formal  and  internal  readjustments  of  expression. 

131.  The  topic  of  rotational  stability  is  also  at  its  core  dynami- 
cal, and  it  is  approachable  most  directly  through  the  considera- 
tions that  we  have  been  attaching  to  regular  precession,  when 
the  possibilities  are  examined  of  securing  that  type  of  adjust- 
ment with  the  (C)  axis  directed  nearly  in  the  upward  vertical. 
We  shall  confine  inquiry,  on  this  side  as  well,  to  outlining  the 
connections;  their  essentials  being  grasped,  the  exhaustive 
treatment  of  details  offers  no  other  obstacles  than  the  inevitable 
mathematical  difiiculties. 

The  first  pertinent  thought  is  derivable  from  equations  (278) 
when  a  range  into  the  second  quadrant  is  permitted  to  (■&), 
and  a  discrimination  needs  to  be  regarded  between  real  and 
imaginary  values  of  the  rotation  about  (Z),  or  between  adjust- 
ments that  can  and  that  cannot  be  accomplished.  Selecting  the 
first  alternative  form  for  the  solution,  this  dividing  hne  is  to  be 
drawn  where  the  values  denoted  here  as  special  yield  the  relation 

0  =  (Caj'(c))2  +  4W?A  cos  t?';        cos  t?'  <  0.       (287) 

And  the  critical  magnitude  which  (w(o))  must  at  least  reach  if 
imaginary  values  are  to  be  excluded  completely  is  given  by 


/4WrA  ,^„^, 

CO(c)    =     ±-y|       ^2       J  ^^°°) 

so  that  if  the  spin  about  (C)  equals  or  exceeds  this  rate,  the 
attainment  of  regular  precession  at  every  position  in  (•&)  is  only  a 
matter  of  providing  the  companion  value  of  the  spin  about  (Z). 
With  this  simple  mathematics  clear  the  next  step  is,  as  in  the 
previous    combination,    to    detect    and    assign    the    dynamical 


176  Fundamental  Equations  of  Dynamics 

reason  that  must  underlie  it.  The  first  stage  in  meeting  that 
requirement  starts  with  the  merely  reshaped  equation 

A^  =sint?(^- Wr-^Ca,(e)  +  A(^^j  cos^j.     (289) 

This  can  be  made  to  tell  us  that  if  the  axis  (C),  having  been 
directed  vertically  upwards,  moves  away  from  that  position,  and 
changes  (t?)  by  a  small  amount  from  the  value  (tt),  it  will  be  true 
that 

(290) 
/      dt?  \  , 
A*  =di(  dr-TT  jdt;        cost?=-l. 

In  words,  the  rotation  rate  (d??/dt)  will  always  be  subject  to 
reduction  in  magnitude  when  the  above  parenthesis  is  itself  a 
negative  quantity;  and  we  have  discovered  a  cause  for  this 
reduction  by  seeing  how  the  weight  moment  meets  a  first  claim 
for  guiding  directional  changes  in  (H);  a  special  case  under 
equation  (286)  is  before  us  now.  The  stronger  such  absorption 
of  force-moment,  the  more  rapid  becomes  that  check  upon  the 
initial  motion  in  (^),  which  will  begin  straightway  as  (C)  leaves 
the  upward  vertical  whenever  the  parenthesis  is  in  the  aggregate 
negative.  Therefore  we  are  led  by  these  considerations  to  look 
at  equation  (284)  in  a  somewhat  new  light  after  rewriting  it 


d^t?  _  sint^ 

dt^"  ~  "  ~  ~    A 


W?  +  ^  Cco(c)  -  A  r  ^  Y  cos  t?  y     (284a) 


Then  a  zero  value  of  the  parenthesis  when  its  factor  is  not  zero 
marks  the  transition  between  favorable  and  unfavorable  con- 
ditions for  checking  an  existing  motion  in  (^).  In  application 
to  the  second  quadrant,  the  third  term  must  be  a  positive  mag- 
nitude always,  but  it  decreases  as  (C)  approaches  a  horizontal 
position.     It  is  clear  that  cases  may  occur  where  the  first  member 


The  Main  Coordinate  Systems  177 

has  unfavorable  sign  as  (r?)  leaves  the  value  (ir),  and  becomes 
favorable  only  after  a  finite  drop  of  the  axis  (C).  Also  it  has 
been  seen  that  the  unfavorable  interval  can  then  be  narrowed 
by  quickening  the  spin  about  (C),  and  it  disappears  at  the 
critical  value  indicated  by  equation  (288).  Because  (sin  t^  =  0) 
is  always  one  solution,  there  is  a  discontinuity  possible  here 
between  the  two  types  of  solution,  similar  to  that  for  the  conical 
pendulum  obtainable  by  assuming  (d<p/dt)  zero  in  the  second 
form  of  equation  (278).  The  classification  sometimes  made  of 
gyroscope  tops  as  weak  and  strong  follows  the  line  of  thought 
just  traced. 

132.  The  factors  in  the  second  term  of  the  parenthesis  that  is 
under  examination  are  never  quite  independent  so  long  as  (d^/dt) 
occurs  in  (w(,.));  but  their  dependence  assumes  a  special  phase 
when  the  (C)  axis  and  the  vertical  can  become  coincident,  for  then 
there  will  be  only  two  different  expressions  for  the  same  (vertical) 
component  of  (H).  In  order  to  develop  the  latter  relation  and 
to  reduce  the  parenthesis  accordingly  we  shall  begin  with  the 
more  general  statement  and  afterwards  particularize  it.  By 
projecting  from  (B')  and  from  (C)  on  the  vertical  and  adding 
we  obtain 


H 


u 


.)   =  tI:i[^^sin7?B'jsinT? 

+  (' ^  +  ^  cos  t?  )  C  cos  t?!.     (291) 
Consequently 
H(,,)  -  t|;i(H(e)  •  <rO  =  tti  (  ^  A  sin^  t?  )  ;         [B'  =  A] ;      (292) 

with  the  general  value  for  the  tensor  ratio 

#  _  H(^,)  -  H(„)  cos  t? 

dt  ~       A(l  -  cos^  ^)      '  ^^^'^''^ 

which  gives  under  the  equality  attendant  upon  coincidence  in 


178  Fundamental  Equations  of  Dynamics 

the  upward  vertical,  the  conventions  for  signs  being  duly  recon- 
ciled, 

Substitution  in  equation  (290)  shows  as  a  condition  that  the 
right-hand  member  should  be  negative  when  (C)  leaves  the 
upward  vertical  with  positive  (At?) 

(Ca>(c))2  >  4AWr.  (294) 

The  greater  this  inequality  the  stronger  the  retardation,  the 
sooner  the  departure  is  brought  to  a  halt.  The  mathematics  of 
equation  (288)  has  found  thus  a  foundation  in  the  dynamical 
process  initiated  when  (C)  leaves  its  vertical  position. 

133.  In  what  precedes,  the  emphasis  falls  upon  moment  of 
momentum  in  relation  to  force-moment.  The  thought  is  not 
complete  however  until  the  work  of  the  weight  moment  has 
been  connected  with  changes  in  kinetic  energy.  For  the  case  in 
hand  we  find  by  using  the  principal  axes, 

[(5¥y+(5f^'°''y]+^c"'<";  (295) 

and  the  last  term  being  constant,  the  variations  or  interchanges 
consequent  upon  work  done  are  confined  to  the  two  other  terms. 
Now  referring  to  equations  (285)  examination  soon  convinces  us 
that  the  initiative,  so  to  speak,  centers  in  the  quantity  that  is  in 
the  fine  of  the  resultant  force-moment.  So  long  as  (dt?/dt)  is 
zero,  no  change  can  occur  in  (co(b'));  but  the  vanishing  of 
(w(b'),  co(c))  separately  or  simultaneously  might  not  prevent 
changes  in  (dt?/dt).  It  is  characteristic  of  the  stability  here  in 
question  that  the  action  depends  vitally  upon  the  actual  oc- 
currence of  a  displacement;  and  this  accounts  for  the  known 
feature  of  gyroscopic  mechanisms,  that  their  efficiency  is  nullified 
by  removing  the  degree  of  freedom  upon  which  their  functioning 
depends. 


E  =  U 


The  Main  Coordinate  Systems  179 

For  the  power  as  the  derivative  of  the  kinetic  energy, we  can  write 

P  =  -4  (Sf  )  5F  +  ""■''  S  ("<'■')  ]  =  M(.-.  5F  •     (296) 

Let  the  conditions  be  such  that  positive  work  is  done,  negative 
moment  being  accompanied  by  negative  displacement.  Then 
the  first  term  in  the  second  member  will  be  negative  for  opposite 
signs  of  its  factors.  And  we  see  diverted  from  their  appearance 
in  the  coordinate  (i?)  the  magnitude  changes  in  both  (H)  and  (E) 
that  (M)  would  make  visible  there,  were  there  no  gyroscopic 
interactions. 

The  general  agreement  of  the  equation  (288)  and  the  inequality 
(294)  in  their  formulation  of  a  critical  value  is  obvious;  and  it 
ought  not  to  be  longer  obscure  why  the  same  truth  is  at  the 
foundation  of  each  criterion.  The  essence  of  the  adjustment  to 
regular  precession  is  the  insufficiency  of  the  available  weight 
moment  at  a  certain  value  of  (t?)  and  other  quantities  to  do  more 
than  supply  exactly  what  is  needed  for  the  corresponding  direc- 
tional change  in  (H).  The  reversal  in  sense  of  the  inequality 
that  we  arrived  at,  declares  in  effect  an  unavoidable  preponder- 
ance of  weight  moment  consistently  with  the  other  given  values, 
and  its  sufficiency  to  quicken  the  motion  in  {■&)  that  is  supposed 
to  exist  already.  It  is  an  easUy  deduced  consequence  therefore 
as  regards  the  axis  (C)  that  it  will  continue  its  departure  from 
the  upward  vertical  until  conditions  alter.  The  imaginary 
range  of  equation  (278)  is  one  signal  that  the  combination  of 
the  accompanying  spin  about  (Z)  with  the  actual  horizontal 
component  of  (H)  is  within  that  region  unequal  to  monopolizing 
the  full  force-moment  active.  The  quantitative  elaboration  of 
these  leading  ideas  produces  the  accepted  results  in  every  detail. 

Generalized  Momenta  and  Forces. 
134.  At  the  date  of  their  original  announcement,  Lagrange's 
coordinates  and  the  equations  of  motion  that  employed  them 


180  Fundamental  Equations  of  Dynamics 

were  contrived  in  the  service  of  what  would  now  be  called 
mechanics  proper,  for  the  imperious  reason  that  the  longer  list 
of  energy  transformations  which  dynamics  distinctively  em- 
braces had  not  yet  been  discovered  and  drawn  into  the  funda- 
mental quantitative  connections.  The  terms  coordinate,  con- 
figuration, velocity  and  momentum  were  enlarged  by  Lagrange 
from  usage  as  he  found  it  no  doubt,  but  his  broader  scheme  did 
not  break  the  alliance  with  geometrical  ideas  for  its  kinematics. 
His  parameters  were  ultimately  based  on  combinations  of 
lengths  and  position  angles,  though  kept  unspecialized  by  sup- 
pressing or  deferring  the  analysis  of  them  into  the  plainer  geo- 
metrical elements.  The  energy  too  was  introduced  primarily  in 
its  kinetic  form,  that  and  momentum  deriving  their  dynamical 
quality  from  those  inertia  factors  that  are  in  their  nature  either 
directly  given  as  mass,  or  else  as  literal  as  moments  of  inertia 
that  emerge  from  a  mass-summation.^ 

Lagrange's  equations  will  be  found  akin  to  Euler's  in  two 
respects:  first  they  are  normally  intended  for  treating  as  a  unit 
some  body  or  system  of  bodies;  and  secondly,  they  are  after  a 
fashion  of  their  own  indifferent  toward  a  substitution  of  one 
system  for  another,  provided  that  determinate  equivalencies  are 
observed,  as  we  have  seen  Euler's  equations  to  be  under  invari- 
ance  of  (A,  B,  C)  in  magnitude.  This  likeness  extends  far 
enough  to  coordinate  the  two  plans  and  to  make  the  latter  when 
duly  stated  a  special  result  of  Lagrange's  broader  handling. 
The  demonstration  offered  by  Lagrange  himself  is  founded  on 
d'Alembert's  principle;  and  this  interconnection  of  the  two 
phases  of  the  same  idea,  and  of  each  with  Hamilton's  different 
formulation  of  it,  lends  to  the  establishment  of  the  equations  of 
motion  an  air  of  logical  redundancy.  This  was  the  subject  of  a 
passing  remark  in  our  Introduction;  and  it  might  be  recalled  too 
that  the  noticeable  swing  away  from  the  first  vogue  of  d'Alem- 

» See  Note  30. 


The  Main  Coordinate  Systems  181 

bert's  statement  centers  upon  a  recent  discovery  of  more  compre- 
hensive adaptability  in  the  alternative  forms  devised  by  Lagrange 
and  by  Hamilton  to  a  range  of  energy  transformations  that  was 
unsuspected  when  either  of  the  latter  was  first  accepted.  By 
the  light  of  what  is  developing  further  in  that  quarter  the  esti- 
mate of  their  fruitfulness  will  continue  to  be  decided. 

Because  these  are  the  origins  it  seems  advisable  to  let  the 
treatment  here  conform  to  them,  instead  of  making  a  short  path 
to  the  newest  reading.  There  is  ground  to  expect  that  the  fuller 
reahzation  of  meaning  in  the  extension  of  method  and  of  its 
valid  possibilities  will  have  its  best  source  in  a  reasoned  apprecia- 
tion of  where  the  latent  power  resided  and  how  it  was  implanted. 
We  hold  one  reliable  clew  already,  wherever  it  proves  true  that  a 
mechanism,  construing  the  word  not  too  remotely  from  direct 
perceptions,  can  be  seen  to  give  in  its  fluxes  of  energy  and  momen- 
tum a  quantitative  equivalent  for  those  fluxes  under  less  restricted 
conditions  of  transformation. 

135.  On  working  outwards  to  occupy  a  broader  field,  and 
passing  at  points  the  hmits  earlier  drawn,  some  elements  of  new 
definition  or  specification  are  involved,  which  the  circumstances 
lead  toward  supplying  in  part  positively,  in  part  by  noting  the 
barriers  that  remain.  And  we  shall  relinquish  the  attempt  to 
finish  each  topic  in  a  systematic  progressive  order,  wherever  it 
promises  better  success  to  proceed  less  rigidly;  coming  back  to 
add  a  stroke  and  explain  or  define  what  was  at  first  only  sketched. 

When  it  is  said  that  any  set  of  coordinates  must  determine  a 
configuration  completely,  the  plain  idea  is  that  they  do  for  a 
system  what  we  expect  of  the  standard  frame  (0,  XYZ),  the 
coordinates  being  enumerated  for  as  many  joints  or  articulations 
as  removal  of  ambiguity  makes  necessary.  If  the  coordinate 
set  is  thus  equivalent  to  (xyz),  the  same  idea  may  be  conveyed 
by  declaring  each  general  coordinate  to  be  a  definite  function  of 
the  set  (xyz).     In  normal  usage  we  do  not  abandon  the  relation 


182  Fundamental  Eqiiations  of  Dynamics 

upheld  for  other  coordinate  systems,  that  the  values  expressed 
with  their  aid  are  standard  frame  values  of  the  quantities  dealt 
with,  but  we  seek  that  aid  through  any  convenient  functions  of 
(xyz)  and  not  merely  through  lines  and  angles.  Such  pre- 
liminary conception  of  a  coordinate  denoted  by  (k)  prepares  the 
way  for  a  definition  of  the  corresponding  velocity  as  (k),  meaning 
the  total  time-derivative  of  the  magnitude  of  (k),  the  question 
about  vector  quality  being  left  open,  an  equal  number  of  veloci- 
ties and  of  coordinates  being  matched  each  to  each. 

Passing  next  to  momentum  we  are  again  confronted  with  a 
definition  that  pairs  each  velocity  with  its  own  momentum 
quantity.  Let  (q)  denote  one  of  these  momenta  belonging  to 
the  velocity  (k);  then  the  defining  equation  is  written,  if  (E) 
is  still  the  total  kinetic  energy  of  the  system  to  be  studied, 

dE 
q^^.  (297) 

And  another  fixed  point  in  the  scheme  now  being  presented  is 
that  (E)  shall  be  a  homogeneous  quadratic  function  of  all  the 
velocities  (k).  To  this  specification  other  things  must  be  made 
to  bend  should  that  become  necessary,  which  is  a  matter  for  due 
inquiry.  But  meanwhile  one  evident  consequence  of  it  can  be 
read  from  the  last  equation,  regarding  the  constitution  of  the 
momenta  (q) ;  they  cannot  be  other  than  linear  functions  of  the 
velocities  (k)  and  homogeneous.  Refer  however  to  the  closing 
remark  of  section  141. 

136.  Putting  together  what  has  been  said,  one  feature  in  the 
relation  of  coordinates  to  configuration  is  caused  to  stand  in 
relief:  they  must  determine  it  in  a  form  free  from  all  reference 
to  velocities  in  order  that  (E)  may  take  on  the  assigned  type. 
Let  us  add  as  being  naturally  required,  that  the  members  of  a 
coordinate  set  must  be  mutually  independent,  and  proceed  to 
speak  of  their  connection  with  the  so-called  degrees  of  freedom 


The  Main  Coordinate  Systems  183 

that  a  system  of  bodies  possesses.  Consideration  of  simplest 
instances,  like  that  of  a  ball  carried  on  the  last  in  a  numerous 
set  of  rods  jointed  together,  shows  that  a  large  number  of  speci- 
fying elements  or  coordinates  may  be  actually  employed  in 
designating  configuration,  even  in  one  plane.  But  we  know  also 
that  two  rectangular  or  two  polar  coordinates  only  are  required 
in  this  case;  and  the  prevaihng  distinction  seems  to  follow  the 
line  thus  indicated,  making  degrees  of  freedom  equal  in  number 
to  the  minimum  group  of  coordinates  requisite  in  describing  a 
configuration,  classing  the  excess  in  the  number  really  used  as 
superfluous  coordinates.  This  disposes  of  the  matter  well  enough, 
leaving  for  special  examination  only  such  interlocking  of  two 
coordinates  into  related  changes  as  happens  when  a  ball  rolls 
(without  sliding)  on  a  table;  and  that  finer  point  need  not  detain 
us.  In  these  terms,  a  rigid  solid  has  available  not  more  than 
six  degrees  of  freedom,  three  of  which  might  call  for  coordinates 
locating  its  center  of  mass,  with  the  remaining  three  covered 
by  the  Euler  angles,  for  example.  And  we  may  borrow  from 
regular  procedure  in  that  case,  as  known  through  repeated 
discussion,  that  an  equation  of  motion  is  associated  with  each 
degree  of  freedom.  That  normal  arrangement  continues  with 
evident  good  reason,  though  our  treatment  is  shaped  according 
to  Lagrange's  proposals,  which  do  not  change  the  objective  in 
essence,  but  only  the  mode  of  reaching  it. 

137.  To  complete  the  plan,  therefore,  into  which  accelerations 
do  not  enter  directly,  there  is  need  to  specify  its  forces;  here  the 
determining  thought  has  its  root  in  the  energy  relations,  running 
in  the  course  that  we  shall  next  lay  out,  whose  first  stage  has  no 
novelty,  but  merely  holds  to  the  equivalence  in  work  established 
for  any  resultant  force.  The  right  to  substitute  one  force  (R) 
for  all  the  distributed  effective  force  elements  depends  upon  its 
equality  with  them  in  respect  to  total  work  and  impulse.  The 
same  thought,  in  other  words,  declares  equal  capacity  for  setting 
13 


184  Fundamental  Equations  of  Dynamics 

up  the  total  flux  of  kinetic  energy  and  momentum  in  relation  to 
the  system  of  bodies,  the  separation  of  force  and  couple  moment 
or  of  translation  and  rotation  being  a  detail  and  without  final  in- 
fluence. It  is  inherent,  moreover,  in  the  determination  of  any 
such  resultant  through  vector  sums  or  through  algebraic  sums 
that  a  set  of  components  may  be  variously  assigned  to  the  same 
resultant.  The  ground  that  Lagrange  traversed  led  him  to  a 
variation  only  on  previous  forms  in  expressing  this  essential 
energetic  equivalence  of  the  resultant  force.  The  fact  indeed 
that  he  set  out  from  the  equilibrium  principle  of  virtual  work  due 
to  d'Alembert  should  obviate  any  surprise  on  meeting  the 
defining  equation  for  his  generalized  forces. 

With  each  degree  of  freedom  which  makes  flux  of  kinetic 
energy  possible,  associate  its  force  (F);  sum  the  work  during 
elements  of  displacement  in  all  the  coordinates  (k)  and  express 
its  necessary  equality  to  the  same  work  given  in  terms  of  the 
usual  forces  parallel  to  (X,  Y,  Z).     The  equation  is 

S(FdK)  =  2/„(dR-ds) 

=  S/„,[dR(,)dx  +  dR(y)dy  +  dR(.)dz],     (298) 

which  yields  by  a  transformation  that  embodies  through  the 
partial  derivative  notation  the  supposition  of  independence  that 
goes  with  the  coordinates,  for  each  force  an  expression 

F  =  2 A  [dRw  ^  +  dR(y)  ^  +  dR(.)  ^  J  .       (299) 

Holding  to  this  statement  any  force  (F)  can  be  defined  in  magni- 
tude by  the  work  per  unit  of  displacement  in  its  coordinate; 
and  the  narrowing  assumption  does  not  appear  that  (F)  and  (k) 
are  colinear,  provided  a  convention  can  be  observed  that  gives 
the  work  its  real  sign  as  determined  by  gain  or  loss  to  the  system's 
kinetic  energy.  It  is  this  relation  which  Lagrange's  equations 
enlarge  by  including  the  other  energies  of  dynamics. 


The  Main  Coordinate  Systems 


185 


We  continue  by  introducing  necessarily  equivalent  expressions 
for  a  change  in  configuration, 


dx  =  2(|dK);         dy  =  z(|dK); 
dz=2(|dK); 


(300) 


in  which  the  summation  extends  to  all  the  coordinates   (k). 
Then  in  the  fluxion  notation 


=  KiO'    ^  =  Ki")'    '-^10' 


(301) 


from  which  follow  for  each  coordinate  singly  the  important 

equaHties 

dx       dx  dy       dy            dz        dz 

dk       dK  '  dk       dK  '         dk       Ok  ' 


(302) 


Taking  the  term  from  the  first  integral  of  equation  (299),  it  can 
be  given  the  form,  by  using  the  last  results 

,^       dx        d  /  ,^       dx\        ,^        d  /dx\ 

dR«  ii  =  S  (<*Q«  ai)  -  -^Q"'  dt  (ii  j '  (^»3^> 

and  similarly  from  the  remaining  integrals, 

^^^->  ^  =  ft  (^^(^>  i)  -  ^^^^^  ft  (i) ' 


dR(z)  —  = 


^  -  A. 

dK  ~  dt 


/  ,^       di'\        ,^       d  ( dz\ 


(303b) 


To  recast  the  last  factors  in  these  three  equations  we  write 


6_ 
dt 


(  d  \       dx  d  [  d  \       dy 

dt\dK^^^  J~  dK' 


(304) 


186  Fundamental  Equations  of  Dynamics 

whose  justification  is  somewhat  a  matter  of  mathematical  con- 
science. The  order  of  the  two  differentiations  may  boldly  be 
inverted  as  a  legitimate  operation;  or  whatever  hazard  may  be 
felt  in  that  can  be  guarded  against  by  rigorous  proofs  that  are 
accessible.  Incorporating  the  last  forms  and  summing  equations 
(303),  the  force  finds  expression  as 


F  = 


-/"[s(^«^'^))-a^(^^)] 


d  fdE\      dE      ,       , 


in  application  to  each  one  of  the  coordinates,  and  the  whole 
development  is  then  open  to  further  comment  or  illustration. 

138.  This  exposition  of  Lagrange's  equations,  and  of  the  con- 
cepts upon  which  their  statement  rests,  has  been  kept  apart 
purposely  from  the  infusion  of  vectorial  ideas,  in  order  to  set 
forth  as  clearly  as  may  be  done  that  possibility  upon  which 
their  larger  usefulness  in  great  measure  depends,  and  of  which 
insistent  mention  was  made  in  the  first  chapter.  Some  care 
seems  needed  to  break  up  the  misleading  connotations  of  words 
like  velocity  and  momentum,  that  in  their  first  and  perhaps 
most  literal  sense  imply  each  an  orienting  vector.  And  the 
■emancipation  of  thought  in  this  regard  has  been  hindered  doubt- 
less by  the  unsuggestive  practice  of  pointing  out  as  examples  of 
this  method  of  attack  solely  those  where  velocities  and  momenta 
and  forces  offer  themselves  habitually  as  vectors — like  those 
which  our  material  has  been  including  hitherto.  If  the  trend 
of  any  demonstration  equivalent  to  the  foregoing  be  watched, 
however,  it  is  seen  to  hinge  essentially  upon  an  enumeration  of  a 
sum  of  terms  in  the  total  energy  of  all  forms  that  are  considered, 
and  analyzing  them  as  products  that  conform  to  a  type.  This 
contains  always  as  a  factor  the  time  rate  of  one  in  a  group  of 
quantities  by  whose  means  the  changes  in  that  energy  content  are 


The  Main  Coordinate  Systems  187 

adequately  determined.  The  success  of  the  analysis  therefore 
depends,  broadly  speaking,  upon  the  isolation  of  suitable  factors 
in  the  physics  of  the  energy  forms  to  specify  the  energy  configur- 
ation and  to  provide  the  necessary  velocities.  And  in  that  direc- 
tion it  is  interesting  to  note  the  part  really  played  by  the  (XYZ) 
velocities  and  momenta  as  they  lead  to  the  vital  connections  in 
equation  (305) .  They  are  scarcely  more  than  a  scaffolding,  an  aid 
in  building  but  removed  from  the  structure  built,  impressing 
effectively  only  one  character  upon  the  result — that  its  scheme  of 
values  shall  be  quantitatively  a  possible  set  in  that  mechanical 
phantom  or  model  which  is  mirrored  in  the  case  treated.  On  their 
face,  Lagrange's  equations  might  seem  to  stand  in  parallel  with 
tangential  ordinary  forces  only,  since  the  latter  are  alone  con- 
cerned in  work.  But  we  shall  show  that  this  limitation  does  not 
in  fact  exist,  and  that  the  pattern  set  by  the  (XYZ)  axes  when 
they  include  for  their  projections  constraints  as  well,  is  stamped 
upon  these  other  combinations,  which  may  be  caused  also  to 
reveal  normal  forces  that  may  be  active  (see  section  141).  As  a 
counterpart  to  this  relation  it  is  to  be  observed  how  the  (XYZ) 
axes  fit  everywhere  into  a  plan  of  algebraic  products  through  their 
three  coexistent  and  practically  scalar  operations;  and  how  for 
the  element  of  scalar  mass  equations  (1,  II)  are  always  free 
alternatives,  whatever  restrictions  subsequent  steps  may  impose, 
as  for  instance  equation  (67)  has  recorded. 

139.  Having  laid  some  preliminary  emphasis  upon  the  extent 
to  which  they  may  exceed  in  scope  other  coordinate  systems,  it 
will  be  advisable  to  carry  the  comparison  with  Lagrange's  plans 
into  the  region  of  overlapping,  and  make  this  last  system  prove 
itself  capable  of  bringing  out  correct  consequences  there  too,  when 
orientation  is  reestablished.  The  cross  relations  have  many 
lessons  that  are  of  value;  and  some  are  yielded  by  a  review  of 
the  polar  coordinates  that  we  shall  put  first.  Borrowing  from 
section  106  the  expression  for  kinetic  energy  of  a  particle,  and 
using  fluxion  notation  for  brevity. 


188 


Fundamental  Equations  of  Dynamics 


E  =  |m[r2  +  r2t?2  +  r^  sin2  ^(^)2]. 


(306) 


The  Lagrange  coordinates  must  be  independent  and  sufficient 
to  give  configuration  in  (XYZ) ;  and  (r,  ^,  i|r)  meet  this  require- 
ment. But  the  velocities  must  correspondingly  be  (r,  i^,  \p). 
The  details  work  out  into  the  forms,  {d'E/dtp)  being  zero, 


dE 
dr 


dE  ■  dE  .        . 

=  mr:        — v-  =  mrt?:        tt  =  nir^  sm^  ^\}/: 


d  /aE\  .  d  /dE\ 

dtUJ  =  -^'        dt(^)=-(2^^^  +  ^^^)' 

d  /aE\ 

-j7  I  T^  )  =  m(2r  sm2  i}4^r  +  2r2  sin  i}  cos  H& 

+  r^sin^t?^^); 


dE 
dr 


=  m(rt?2-|-rsin2i?(^))2; 


dE 


=  m(r2  sin  t?  cos  i>(^))2. 


(307) 


A  general  agreement  is  at  once  manifest  when  these  terms  are 
grouped  and  compared  with  equations  (208) ;  but  it  is  a  striking 
difference  that  the  forces  (F(^))  and  (F(^)),  associated  with 
those  two  coordinates,  must  now  be  recognized  as  moments  of 
the  forces  denoted  previously  by  (R(x'))  and  (R(y')),  for  rotation- 
axes  characterized  plainly  through  the  respective  lever  arms. 
This  is  a  necessary  concomitant  of  making  velocities  out  of 
(t?,  rp).  The  regrouping  of  terms  also  is  instructive  in  betraying 
that  loss  of  distinction  for  the  orientation  changes  here  as  well 
which  algebra  usually  evinces. 

140.  For  a  second  example,  let  us  make  in  the  Lagrange  form 
a  restatement  of  section  89,  utilizing  equations  (154)  as  a  starting- 
point,  and  adapting  them  to  a  particle,  as  the  desirably  simple 
case.  If  (x',  y',  z')  are  selected  as  three  coordinates,  the  con- 
figuration in  (XYZ)  is  not  determinate  by  them  alone,  but  in 
the  plan  followed  the  position  angles  for  the  axes  (X'Y'Z')  must 
be  known  also;  and  of  these  as  many  as  are  independent  can  be 


The  Main  Coordinate  Systems 


189 


added  to  make  the  required  list  of  coordinates,  of  which  all  but 
three  will  then  be  superfluous  in  a  sense  already  explained,  and 
not  to  be  reckoned  among  the  degrees  of  freedom.  The  purpose 
of  illustration  can  be  attained  sufficiently  if  we  consider  the 
uniplanar  conditions,  both  for  the  particle  which  is  then  supposed 
to  be  restricted  to  the  (XY)  plane,  and  for  the  relative  con- 
figuration of  (X'Y'Z'),  where  we  assume  (Z)  and  (Z')  permanently 
coincident.  Hence  for  the  kinetic  energy  of  (m)  the  expression 
is  in  understandable  terms 

E  =  |(x2  +  y2)m  =  im[x'2  +  y'^  +  (x'^  +  y")y' 

-  2xy7  +  2x'y'i],     (308) 

the  coordinates  being  now  (x',  y',  7)  and  the  velocities  (x',  y',  7) ; 
the  last  velocity  is  an  algebraic  derivative,  (Z)  being  the  fixed 
axis  for  (y).  Again  the  details  are,  when  this  homogeneous 
quadratic  function  of  the  velocities  is  differentiated. 


^,  =  m(x    -y7); 


-^,  =  m(y   +  X  7) ; 


dE 

dy 

dE 
dx' 

dE 

or 


=  m{y{x''  +  y')  -xy +  x'y'); 
=  mij^x'  +  7y'); 
=  m(7y  -  7x'); 


dE 
dy 


0. 


(309) 


After  forming  the  time-derivatives  of  the  first  three  in  the  group 
and  substituting  values,  we  obtain  for  the  three  forces  of  the 
coordinates, 

F(.')  =  m(k'  -  7y'  -  2iy'  -  7V); 

F(y')  =  m(y'  +  7x'  +  27i'  -  7y);  ^  (310) 


(v) 


=  x'F(,')  -  y'F 


(x)' 


1 90  Fundamental  Equations  of  Dynamics 

The  third  coordinate  advertises  that  it  is  superfluous,  in  that  its 
force  value,  whose  form  is  readily  verifiable  as  a  moment,  only 
confirms  what  is  otherwise  ascertained  about  the  remaining 
forces. 

141.  In  their  adaptation  to  the  present  class  of  cases,  some 
truths  can  be  picked  out  that  furnish  clews  for  the  lines  of  more 
extended  use.  First,  referring  to  equations  (155)  and  collating 
them  with  equations  (302,  304),  the  latter  are  seen  to  be  far- 
reaching  analogues  of  changes  that  build  upon  the  line  of  the 
quantity  at  the  epoch,  and  of  those  others  that  depend  upon  a 
change  of  slope;  they  are  correlated  respectively  with  changing 
tensor  and  orientation  of  a  vector.  While  a  partial  derivative 
like  (dx/dK)  may  appear  as  a  direction  cosine  within  the  purely 
geometrical  conditions,  it  is  a  more  inclusive  reduction  factor  else- 
where. It  is  also  open  to  observation  in  the  last  two  illustrations 
that  the  generalized  momenta  become  for  those  applications  the 
orthogonal  projections  upon  a  distinguishable  line,  either  of  the 
momentum  or  of  the  moment  of  momentum  in  the  standard 
frame.  Differences  of  distribution  for  the  same  total  projection 
between  various  pairs  of  groups  is  no  more  than  part  of  the 
mathematical  machinery,  and  it  is  especially  to  be  expected 
where  sets  of  partial  derivatives  occur  whose  variables  have  been 
changed.     Note  that 

51'    ^'  (^"> 

presuppose:  the  first,  that  all  coordinates  are  held  stationary* 
and  all  velocities  but  that  one;  and  the  second,  that  only  the 
one  coordinate  is  allowed  to  change,  and  none  of  the  velocities. 
Comparisons  with  other  sets  of  partials  in  our  developments 
should  prove  helpful,  as  it  will  be  to  find  answer  for  the  question 
whether  the  Lagrange  plan,  when  it  deals  with  forces  like  (R), 
affiliates  more  closely  with  the  mode  of  equation  (112)  or  with 
that  of  equation  (233). 


The  Main  Coordinate  Systems  191 

Related  to  the  second  example  here  and  to  the  ideas  about 
superfluous  coordinates,  is  another  point  of  view  that  has  like- 
ness with  the  method  of  section  82.  The  standard  frame  coordi- 
nates, as  expressed  in  equations  (150),  can  be  discriminatingly- 
dependent  upon  time,  indirectly  through  (x',  y',  z')  and  directly 
through  the  direction  cosines.  Their  exact  differentials  will  then 
appear  as 

3x    ,   ,       dx    ,   ,       dx    ,  ,       3x  ,  ,^  ^^ 

with  two  companions,  the  last  term  in  each  comprising  the  group 
that  arise  by  differentiating  the  direction  cosines  if  we  have  re- 
garded (xyz)  as  given  in  a  functional  form  Uke 

X  =  f(x',  y',  z',  t),  (313) 

and  the  superfluous  elements  are  spoken  of  and  dealt  with  as 
due  to  variations  of  the  geometrical  relations  with  time.  The 
distinction  that  such  changes  of  direction  are  assigned  and  not 
brought  about  by  physical  action  is  consistent  with  what  has 
been  seen  above — the  absence  of  those  additional  force  speci- 
fication^ that  would  be  introduced  through  them  otherwise. 
The  exercise  of  preference  in  selecting  the  elements  to  be  drawn 
off  thus  into  their  own  time  function,  however,  need  not  be  always 
the  plainest  of  matters.  And  where  an  accompanying  verbal 
usage  is  accepted  that  denies  the  title  coordinate  to  position 
variables  not  ranked  among  degrees  of  freedom,  the  kinetic 
energy  ceases  to  be  a  homogeneous  quadratic  function  of  the 
(remaining)  legalized  velocities.  Of  course  these  comments 
hold  good  for  extension  to  the  generalized  energy  configuration. 

142.  Retaining  the  energy  value  and  imposing  upon  equations 
(310)  the  conditions  that  (7)  and  the  origin  shall  be  so  regulated  as 
to  keep  (V(y'))  at  zero  permanently,  they  conform  to  the  tangent 
and  normal  resolution  of  force  for  those  uniplanar  restrictions; 
and  in  space  curves  there  is  the  same  correspondence  between 


192  Fundamental  Equations  of  Dynamics 

the  general  case  and  the  one  duly  specialized.  The  test  of  the 
latter  form  being  of  some  length  and  of  no  difficulty,  and  because 
it  shows  finally  only  an  equivalent  for  section  115,  we  pass  it 
with  mention  merely  and  proceed  to  examine  Euler's  equations 
for  instructive  connections  with  those  of  Lagrange. 

We  can  quote  two  equally  valid  expressions  for  rotational 
energy  of  a  rigid  solid  for  which  (A  =  B),  when  mounted  as  in 
section  127: 

=  h(^  +  (^  sin  t?)2)A  +  K^  +  'A  cos  i}yC.     (314) 

In  the  former,  no  total  time-derivatives  can  be  detected  of  quan- 
tities determining  configuration,  but  only  those  projections  of  a 
given  (w)  appear  which  presuppose  knowledge  of  the  configura- 
tion, and  which  could  be  rated  partial  derivatives  of  (y)  accord- 
ing to  the  explanation  of  equation  (185)  as  related  to  section  79. 
This  fact  has  been  noticed  in  several  connections  since  the  subject 
of  position  angles  was  opened  (see  sections  93  and  98),  and  it 
explains  why  the  direct  expression  by  means  of  the  Euler  angles 
is  not  entirely  superseded  by  using  (<.>(»),  "(b),  w(c)).  The  co- 
ordinates are  then  (t|r,  ^,  ^),  the  velocities  (^,  t?,  ^)  in  the 
fluxion  notation,  and  we  foresee  that  our  previous  force-moments 
will  now  figure  as  forces.     It  is  plain  that 

-  =  _  =  0;        F<„  =  F<„  =  0;  (315) 

the  latter  pair  of  values  expressing  the  controlling  constancies  of 
the  moment  of  momentum  in  this  problem,  or  of  the  momenta 
(q(^),  q(^))  in  the  present  terminology.  These  values  when 
worked  out,  and  those  that  complete  the  expression 

=  -Ji  (q(^))  -  IT  ,  (316) 


w 


dt'^^'^'^^       d& 


are  all  in  recognizable  identity  with  what  was  obtained  elsewhere. 


The  Main  Coordinate  Systems  193 

143.  The  action  of  the  gj'^roscope  has  been  seen  capable  of 
diverting  energy  from  one  coordinate  to  another  as  a  perhaps 
secondary  consequence  of  maintaining  change  of  direction  in  a 
moment  of  momentum  that  is  of  constant  magnitude.  And  it  is 
easy  to  multiply  instances,  wherever  the  inertia  factors  (moments 
and  products  of  inertia)  can  be  variable,  that  a  change  in  value 
for  kinetic  energy  is  demanded  under  constancy  of  the  other 
quantity,  this  being  entailed  if  the  rotation  factor  alters.  Thus  a 
synometrically  shrinking  homogeneous  sphere  has  constant  (H) 
under  the  influence  of  gravitational  self -attractive  forces  between 
its  parts,  but  the  rotational  energy  grows  as  an  expression  of 
work  done  in  the  shortening  lines  of  stress.  In  symbols,  for 
rotation  about  a  diameter, 

H  =  o>I(„) ;         E  =  ico^Ic^)  =  i  (  j^  )'  I(o)  =  ^-- ,     (317) 


21 


(D) 


with  the  denominator  growing  continually  smaller.  What  is  here 
illustrated  is  more  widely  possible  to  happen  among  the  analogous 
factors  of  energy,  where  its  different  forms  are  interconnected  in 
the  same  system,  so  that  the  energy  may  be  transferred  and 
redistributed  among  the  Lagrange  coordinates  though  some  of  the 
corresponding  momenta  remain  unaltered.  Neither  is  it  remote 
from  the  mental  attitude  already  alluded  to,  in  approaching  the 
study  of  a  physical  system  through  certain  external  and  accessible 
bearings  of  it  while  a  margin  is  left  for  less  definite  inference,  to 
base  tentative  conclusions  about  concealed  constant  momenta 
upon  observable  indirect  effects  on  energy.  It  is  some  prepara- 
tion for  those  fields  of  usefulness  to  follow  out  the  relations  in  the 
next  sequence  of  ideas,  which  may  be  carried  through  first  for 
directed  momenta  and  finally  be  restated  more  broadly. 

We  shall  suppose  a  system  with  four  generalized  coordinates, 
three  (^,  r?,  <p)  what  we  have  termed  accessible,  and  details  about 
the  fourth  (r)  to  be  subjects  for  inference,  as  we  may  say.     The 


194  Fundamental  Equations  of  Dynamics 

latter  has  then  naturally  no  force  assigned  to  it  for  direct  con- 
nection with  changes  of  energy,  and  is  adapted  to  the  thought 
expressed  above,  by  having  its  momentum  assumed  a  constant 
magnitude.     Accordingly  these  conditions  are  written 

^M  —  Oj         QCt)  =  constant.  (318) 

Add  the  supposition  as  conforming  reasonably  to  the  limitations 
upon  knowledge,  that  no  known  relations  contain  (r)  itself. 
Then  since 

d  .       .       dE 


(t) 


=  dt(''<">-ir'  (^i«) 


each  term  in  the  second  member  vanishes  separately  or  is  a  blank. 
144.  The  momentum  (qcr))  being  actually  present  can  modify 
the  phenomena;  that  is  the  effects  of  other  forces  and  the  energy 
reactions.  It  is  to  be  asked :  How  will  the  statements  be  recast, 
if  we  detect  (q(T))  as  though  distributed  in  parts  added  to  the  other 
momenta,  to  which  the  phenomena  are  being  exclusively  ascribed? 
This  moves  in  the  direction  of  suspending  direct  inquiry  into  (t), 
so  the  method  is  frequently  described  as  allowing  ignoration  of 
coordinates.^  Expressing  this  resolution  of  (q(T))  with  the  aid  of 
the  direction  cosines  (1,  m,  n),  and  adding  its  components  to  the 
other  momenta  as  indicated,  the  total  orthogonal  projections  on 
the  lines  will  indicate 

aE         ,  ,  dE         , 

jT  =  q  (*)  +  iq(r);      ^  =  q  (-»  +  ^^m; 
^  SE  (^20) 

^  =  q'(4.)  +  m^r)' 

The  coordinates  {\[/,  r?,  </>)  need  not  be  themselves  orthogonal,  but 
the  parts  (q')  and  (qcr))  are. 

The  adjudged  energy  (E)  would  then  have  to  satisfy  the  general 
relation  growing  by  implication  out  of  the  real  scalar  product  for 
rotation 

»SeeNote31. 


The  Main  Coordinate  Systems 


195 


E  =  Kto-H),  (321) 

the  possible  non-linearity  of  any  velocity  (k)  and  its  momentum 
(q)  being  here  also  recognized;  this  yields  the  form 

E  =  UHq'w  +  lq(r))  +  Hq'w  +  mq(,)) 

+  ^(q'(^)  +  nqco)].     (322) 

Introducing  (Q)  in  this  connection  to  denote  the  constant  magni- 
tude (q(T)),  the  forces  derivable  from  the  supposed  energy  will 
appear  as  containing  the  terms 


d  /aE\  ^  d  /^(o)\       ^dl 
ityd^p  J      dt  V    04^'  J  "^  ^dt' 

/clE\  _  d  /aE(o)\  dm 

/aE\_  d  /aE(o)\ 


dt 

d_ 
dt 

d_ 

dt 


dt  ' 

dn 
dt  • 


(323) 


The  quantity  of  energy  (E(o))  represents  what  would  be  present 
if  (Q)  were  non-existent,  and  the  last  terms  in  the  equations 
register  the  modification  due  to  the  introduction  of  (Q)  on  the 
supposed  basis,  namely  through  its  resolved  parts  that  maintain 
the  directions  of  the  momenta  (q(j,),  q(,>),  q(*))-  Their  indi- 
cated connection  with  changes  of  direction  relative  to  (^,  t?,  <p) 
momenta  should  not  pass  unnoticed.  To  conform  with  the  above 
values,  the  energy  (E')  allowed  for  in  excess  of  (E(o))  must  be 

E'  =  ^(IQ)  +  ^(mQ)  +  ^(nQ);  (324) 


and  in  order  to  fill  out  consistently  the  scheme  begun  with  equa- 
tions (323)  we  must  continue  in  the  expressions  of  force  with 

dE       3E(o)       ^E  dtj 

aE 

dtp 


dxl^ 


dE(o)       dE 

dd   "*"  a^ 


aE(o)     aE 

dtp  .        dtp 


(325) 


196 

But  we  find 


Fundamental  Equations  of  Dynamics 


dE'       ^  /  .  al        .  dm         dn  \ 


dE 
dE 


dyp 


dm 


d^J  ' 


E'      ^f  ,  d\        .dm         dn\ 

(p  \     ocp  dtp  otp  J 


(326) 


Hence  the  aggregate  departures  from  the  forces  that  would  be 
indicated  by  (E(o))  alone  can  be  seen  in 


dE 


dt\dip  )      d^p 


d_/aE(o) 
dt\     d^|y' 


\_dE, 

J  drP 

d_/aE\  _  dE       d  /aE(o)\  _  dE(o) 
dt  V  5??  y      a??  ~  dt  V    d^    )  ~    dt? 


.  ai      .am 

^a^  +  ^a^+^^ 


^%)]' 


+  Q 


Ldt  "V 


.  .  ai      .am 


^^)]' 


d_/aE\  _aE  _  d  /aE(o)\  _  a.E(o) 

/       d(p       dt  \    a^    /  d(p 


dt  \d<p 


d(p     dt  \  a 

^  r  dn     /  .  ai        am        an  \  1 


(327) 


145.  But  the  energy  really  introduced  by  the  momentum  (Q), 
like  the  other  portion  E(o)  of  the  energy  is  expressible  by  a 
homogeneous  quadratic  function  of  the  velocities  which  it  is 
permissible  at  any  one  epoch  to  put  into  the  form 

E(Q)  =  iK(;  +  1.A  +  m#+  n^)2,  (328) 

(K)  being  a  function  of  coordinates  only,  and  the  value  being  in 


The  Main  Coordinate  Systems  197 

other  respects  fixed  by  necessary  relations  for  partial  derivatives 
of  E(Q).     Thus 


aE(Q) 

dr 


=  K(t  -\- U  +  m4  +  n<p)  =  Q  [by  definition]; 


aE(Q)  aE(Q)  aE(Q) 

[by  equations  (320)]. 


(329) 


Further  we  have,  since  (E(q))  involves  coordinates  through  both 
factors, 


?|ai  =  ig(,  +  ,^ +  ,,  +  „,). 


,  f  .  dl  dm  dn\ 

+  K(r  +  l^  +  mt?  +  n^)(^^-  +  ^^+^^j; 


(330) 


and  the  second  part  is  recognizable  through  equations  (327, 
329).  In  order  to  adapt  the  remaining  part  to  the  present 
connection,  first  put  equation  (328)  into  the  legitimate  form 
next  shown,  and  then  express  its  partial  derivative  for  a  coordi- 
nate, subject  to  our  condition  that  (Q)  is  a  constant  magnitude. 
The  results  are 


E(Q)  -  2  K 

aE(Q) 

iQ^aK        laK 

(331) 
(;  +  1^  +  m,?  +  n^),2 

and  the  last  member  is  identified  as  the  negative  of  the  corre- 
sponding quantity  in  equation  (330) .  Its  appearance  in  the  final 
forms  is  intimately  related  to  a  diversion  of  energy  that  persists, 
though  the  action  of  (Q)  is  veiled  otherwise.  Utihzing  all  these 
detailed  relations  justifies  the  equality,  where  the  notation  for 
the  last  term  in  the  first  member  indicates  the  condition  observed, 
and  for  (dl/dt)  we  have  inserted  the  value 


198 


Fundamental  Equations  of  Dynamics 


dl       d\    .       d\    .       d\ 


d  /aE(o)\       aE(o)       qP/^I 


^"^   ^    (332a) 


—  T7(E(o)  +  E(Q))   -  F(j,). 


To  which  the  companions  added  after  cycHc  interchanges  are 
"^  \d<p        a??  /  ^  J  "^  L     5^     Jq       dt\d4  J 


—  ^  (E{o)  +  E(Q))  —  F(^); 


^/aE(o^\ 

dt  \    d(p    J 


dE 


(0) 


d(p 


+  Q 


+ 


Vdt?       5^  /     J        L    ^<^    Jq       dt\d<p  / 


dE 


—  ^  (E(o)  +  E(Q))  =  F(^). 


(332b) 


146.  It  is  plain  from  these  forms  how  the  actual  values  of  the 
last  members  but  one  for  the  energy  changes  in  the  system  may 


The  Main  Coordinate  Systems  199 

be  preserved  and  an  account  of  them  be  given  under  various 
other  interpretations  that  are  in  a  sense  fictitious.  Or  they  are 
put  in  a  fashion  that  uses  knowledge  up  to  its  borders,  with  safe 
non-committal  beyond  them.  What  is  here  exemplified  for  one 
coordinate  ignored,  can  be  extended  of  course  to  many  by  a 
similar  procedure.  And  when  acceptance  of  reduction  factors 
has  widened  the  range  outside  that  covered  by  the  geometrical 
direction  cosines,  intricacies  of  energy  connections  are  made 
resolvable  in  many  general  ways. 

It  may  happen  that  some  contributions  to  the  total  group  of 
forces  acting  on  a  system  are  comprised  under  a  potential  energy 
function;  and  it  is  in  the  nature  of  those  relations  that  such 
forces  are  independent  of  velocities.  If  therefore  there  is  any 
gain  in  doing  so,  the  active  forces  may  be  held  asunder  in  two 
groups,  one  containing  all  the  forces  derivable  from  any  potential 
energy  functions  ($).  Then  in  any  coordinate  (k)  the  new  model 
of  Lagrange's  equations  is  only  formally  varied  when  it  is  written 


^''-  =  fM^^-^^)-h^^-*^' 


(333) 


since  ($)  is  inoperative  in  the  first  term,  and  in  the  second  it 
only  transposes  one  group  of  the  forces.  But  this  type  offers 
the  significant  feature  that  a  course  of  events  to  which  the  first 
member  can  be  the  key,  is  exhibited  as  depending  upon  the 
momentary  outstanding  difference  between  two  quantities 
measurable  as  energy.  And  with  the  door  opened  as  usual  to 
seemingly  vital  analogies  among  energj^  forms,  much  is  being 
done  in  these  days  to  increase  the  command  of  dynamical  state- 
ment for  the  most  inclusive  rules  or  principles  deciphered  among 
physical  sequences  of  transformed  energy.  It  did  not  seem,  there- 
fore, that  the  objects  of  the  chapter  on  the  side  of  stimulating 
suggestion  would  be  attained  unless  we  were  brought  to  this  gate- 
way into  a  larger  field.  But  then  too  we  must  be  content  with 
14 


200  Fundamental  Equations  of  Dynamics 

that  much  of  accomphshment,  leaving  the  other  forms  of  La- 
grange's equations,  beside  this  second  one  as  they  are  usually 
counted,  to  the  systematic  continuations  of  which  there  is  no  lack. 
The  exploitation  of  the  concept  called  kinetic  potential,  whose 
roots  can  be  traced  in  the  difference  (E  —  <^),  and  its  alternative 
origin  as  a  deduction  from  Hamilton's  principle  of  stationary 
action,  are  the  groundwork  of  much  modern  dynamical  thought.^ 

1  See  Note  32. 


Notes  to  Chapters  I-IV. 

Note  1  (page  2).  To  be  aware  of  are  an  initial  trend  through 
the  drift  impressed  by  the  nature  of  the  material,  as  well  as  an 
active  later  movement  with  its  propaganda.  Regarding  the 
first  of  these  headings  it  is  discussible  whether  the  opinion  alluded 
to  in  section  3  is  fully  representative  of  Newton's  own  stand- 
point, or  whether  that  tendency  to  one-sided  development  was 
due  to  adherents  whose  acceptance  of  ideas  was  narrower  than 
the  scheme  of  his  proposal.  So  much  can  be  done  by  way  of 
expanding  or  contracting  the  thought  lying  behind  a  condensed 
formulation  in  Latin  that  we  tread  on  insecure  ground  in  at- 
tempting a  decision.  Safest  it  seems  to  allow  in  Newton's 
plan  at  least  potential  provision  inclusive  of  all  that  two  succeed- 
ing centuries  could  reasonably  urge  on  this  score.  Adding  per- 
haps, what  expert  judges  would  have  us  not  overlook,  that  a 
comprehensive  power-equation  is  laid  down  in  the  scholium  to 
the  third  law.  Read  in  English  thus:  "If  the  Activity  of  an 
agent  be  measured  by  its  amount  and  its  velocity  conjointly; 
and  if,  similarly,  the  Counter-activity  of  the  resistance  be  meas- 
ured by  the  velocities  of  its  several  parts  and  their  several 
amounts  conjointly,  whether  these  arise  from  friction,  cohesion, 
weight  or  acceleration; — Activity  and  Counter-activity,  in  all 
combinations  of  machines,  will  be  equal  and  opposite"  (Thomson 
and  Tait,  Natural  Philosophy  (1879),  Part  I,  page  247).  The 
genius  of  Heaviside  for  directest  dynamical  thinking  approves 
this  scholium  as  capable  of  covering  the  fluxes  and  transforma- 
tions of  energy  that  more  recent  dynamics  introduces  (Electro- 
magnetic Theory,  III,  pages  178-80). 

In  the  movement  toward  basing  the  derivation  of  other  con- 
cepts upon  energy,  Tait  put  forward  an  early  denial  of  primary 

201 


202  Fundamental  Equations  of  Dynamics 

quality  to  force  in  a  lecture  before  the  British  Association  (1876). 
The  habits  of  thought  in  these  respects,  however,  are  interwoven 
with  a  widespread  campaign  extending  over  the  main  issues  of 
epistemology  (Erkenntnistheorie)  that  enlivened  the  period  1895- 
1905,  some  of  whose  other  aspects  are  touched  upon  subsequently 
(see  notes  4  and  5).  The  party  there  whose  watchword  was 
"Phenomenology"  made  common  cause  with  energetics  as  a 
properly  neutral  mode  of  statement,  in  opposition  to  theoretical 
physics — or  more  justly  to  overweight  in  speculation.  These 
matters  of  broad  sweep  are  only  to  be  hinted  here;  they  are  fully 
in  evidence  throughout  the  journals  of  that  date.  Yet  we  may 
admit  mention  of  two  books,  one  showing  how  energetics 
counterpoises  and  supplements  other  aspects  of  dynamics,  and 
the  second  exhibiting  by  contrast  exaggerations  into  which 
zealous  advocates  were  led.  The  titles  are :  Helm,  die  Energetik 
(1898);  Ostwald,  die  Naturphilosophie  (1902). 

Note  2  (page  4).  The  spirit  of  this  paragraph  finds  confirma- 
tion in  recent  judicial  utterances,  as  regards  both  appreciation 
of  the  new  movement  and  prudent  reserve  in  passing  judgment. 
Consult  Silberstein,  The  Theory  of  Relativity,  for  a  lucid  account 
of  the  Lorentz-Einstein  method  that  estimates  its  gains  with 
candor  and  acumen.  The  workable  value  in  the  opened  vein 
of  possibihties  will  be  extracted  progressively,  as  its  logic  is 
brought  to  bear  upon  questions  involving  previous  sequences 
and  their  origins.  Poincar^  expresses  this  plainly  in  his  summing 
up:  "Aujourd'hui  certains  physiciens  veulent  adopter  une  con- 
vention nouvelle  ...  plus  commode,  voil^  tout.  .  .  .  Ceux  qui 
ne  sont  pas  de  cet  avis  peuvent  l^gitimement  conserver  I'an- 
cienne.  .  .  .  Je  crois,  entre  nous,  que  c'est  ce  qu'ils  feront  encore 
longtemps"  (Dernieres  Pensees,  page  54).  Clarification  and 
settlement  here  seem  delayed  by  an  observable  tendency  to 
expound  the  central  ideas  of  relativity  in  an  entanglement  with 
much  irrelevant  mathematics  that  is  describable  also  as  tran- 


Notes  to  Chapters  I-IV  203 

scendental.  This  blurs  essentials  and  will  obstruct  the  final 
rating  of  the  novel  features  among  the  resources  of  physics. 
It  is  foreign  to  such  alhance,  and  hence  perhaps  one  influence 
toward  dissolving  it,  that  the  modified  handUng  of  simultaneous- 
ness  traces  its  Uneage  so  directly  to  experimental  evidence,  and 
the  effort  to  state  its  results  with  unforced  symmetry.  Yet  on 
that  side,  too,  there  might  arise  need  of  corrective,  if  perchance 
the  conclusion  were  entertained  seriously,  that  any  newly  as- 
sumed attitude  releases  us  from  that  bondage  to  idealized 
concepts  and  simpKfying  approximations  which  sections  12  and 
13  indicate.  We  should  be  compelled  to  reject  every  inference 
that  some  system  invented  to  replace  Newtonian  dynamics  can 
be  other  than  differently  conceptual  and  approximate.  What 
alternative  concepts  to  employ  will  always  remain  as  a  choice 
determined  on  practical  grounds.  It  would  be  breaking  with 
the  canons  of  sound  scientific  doctrine  to  displace  one  series  of 
working  ideas  by  another  whose  improved  adaptation  to  universal 
service  is  at  best  to  be  classed  among  open  questions.  Though 
symmetry  in  equations  is  desirable,  it  is  not  to  be  secured  at  all 
costs.  In  order  to  turn  the  balance  conclusively,  insight  must 
first  be  attained  that  goes  far  enough  in  excluding  illusion  from 
the  corresponding  dynamics.  The  characteristic  formulas  of 
relativity  draw  their  suggestion  from  groups  of  phenomena  that 
spread  over  limited  area  as  compared  with  the  explored  range 
of  physics.  Their  analysis  beyond  the  kinematical  stage,  more- 
over, is  too  obscure  and  intricate  as  yet  to  afford  mandatory 
reasons,  or  even  trustworthy  guidance,  for  much  reshaping  of  our 
fundamental  equations.  See  note  11  below,  in  continuance  of 
this  thought. 

Note  3  (page  6).  The  reference  is  to  Maxwell's  Treatise  on 
Electricity  and  Magnetism,  II,  Chapters  V  and  VI  of  Part  IV. 
He  records  (1873)  the  stimulus  received  from  the  Natural 
Philosophy  by  Thomson  and  Tait  (1867),  and  from  the  revival 


204  Fundamental  Equations  of  Dynamics 

of  dynamical  advance  inspired  by  "  that  stiff  but  thoroughgoing 
work"  (Heaviside).  It  continues  to  offer  an  unexhausted  mine 
to  a  later  generation.  In  its  second  edition  (1879)  the  present 
topic  by  added  material  and  recasting  points  rather  plainly 
toward  mutual  reaction  between  Maxwell  and  its  authors.  It 
is  true  that  their  expanded  treatment  does  not  explicitly  occupy 
his  larger  field,  though  their  gyroscopic  illustrations  run  easily, 
as  can  be  seen,  into  a  generalized  scheme  of  cyclic  systems.  In 
that  direction  Ebert,  with  Chapters  XX-XXII  of  his  Magne- 
tische  Kraftf elder  (1897),  has  made  a  junction  by  elaborating 
into  dilution  the  results  of  Hertz  and  Helmholtz.  Others  like 
Gray  prolong  directly  the  line  of  Maxwell's  initiative  (Absolute 
Measurements  in  Electricity  and  Magnetism,  II,  Part  I,  Chapter 
IV  (1893)). 

It  is  not  premature  to  remark,  in  anticipation  of  notes  30 
and  31,  and  with  bearing  upon  the  current  presentations  of 
Lagrange's  equations,  how  guardedly  the  vectorial  connections 
of  their  original  scope  are  relaxing.  We  may  suppose  that  the 
freedom  to  cut  loose  in  this  respect  has  been  for  a  time  masked 
by  the  cartesian  (XYZ)  forms,  whose  effective  reduction  to 
quasi-scalar  expressions  has  had  an  influence  elsewhere,  as  pointed 
out  in  section  91,  toward  indifference  about  such  distinctions 
that  fails  to  regard  them  as  vital. 

Note  4  (page  9).  What  is  appropriate  here  in  preparing  for 
intelligent  command  of  stock  resources  must  not  go  far  beyond 
claiming  for  these  inquiries  a  continued  relation  to  the  organic 
structure  of  dynamics  of  which  their  perennial  life  is  one  con- 
vincing proof.  Some  study  of  their  literature  cannot  be  dis- 
pensed with,  from  which  differently  shaded  opinions  will  be 
drawn,  to  be  sure,  that  will  yet  unite  in  agreement  on  the  final 
importance  of  the  answers.  To  recommend  this  as  one  region 
for  deliberate  thinking  is  the  purpose  at  this  place,  leaving 
opinions  to  shape  themselves  individually.     The  concession  how 


Notes  to  Chapters  I-IV  205 

fully  routine  belonging  to  execution  can  go  its  way  unhampered 
by  deeper  questions  should  be  permitted  to  repeat  itself  without 
undermining  finally  the  need  incumbent  upon  us  to  discuss  them. 
Section  16  alludes  to  some  temporary  grounds  for  unconcern, 
others  are  supplied  by  the  sufficiency  of  a  fixed  earth's  surface 
for  staging  so  many  investigations  of  physics,  and  in  various 
directions  a  fortunate  postponement  is  tolerated.  But  testimony 
is  broadcast  how  steadfastly  some  settlement  is  nevertheless 
held  in  view,  for  the  experimental  bearings  of  it  even,  when  freed 
from  all  metaphysical  residue.  For  exemplifying  reference  take 
Larmor's  comment  (Aether  and  Matter,  page  273)  and  Helm's 
pertinent  remark  (Energetik,  page  216). 

There  were  several  leaders  in  the  public  sifting  of  these  theories. 
Prominent  among  them  Mach,  who  has  gone  on  record  in  his 
Science  of  Mechanics,  Chapter  II,  and  elsewhere.*  The  possi- 
bility of  the  so-called  Newtonian  transformation  having  been 
put  on  a  secure  basis,  that  headed  unconstrainedly  toward  using 
an  origin  at  the  center  of  mass  of  the  solar  system  and  directions 
determined  by  the  stars  for  a  natural  reference-frame.  Espe- 
cially for  what  are  rightfully  classed  as  internal  energies  of  the 
system  this  would  be  capable  of  high  precision  in  presenting 
through  accelerations  relative  to  it,  for  the  bodies  with  which 
we  deal,  the  physical  forces  active  among  them  or  upon  them 
(see  section  52,  and  note  17).  It  is  a  live  question  of  the  passing 
time  whether  that  habit  of  mind  had  better  be  upset,  or  can  be 
superseded  with  definite  net  gain. 

Note  5  (page  17).  The  assertion  is  hardly  contestable,  that 
quantitative  physics  deals  with  an  idealized  and  simplified 
skeleton  built  of  concepts,  so  soon  as  its  content  exceeds  the 
rules  that  are  empirical  by  intention  and  form.  The  supports 
found  for  outstanding  argument  are  then  two:  first,  uncom- 

*  This  is  the  briefer  title  of  the  English  translation,  the  original  title  being 
"Die  Mechanik  in  ihrer  Entwickelung  historisch-kritisch  dargestellt." 


206  Fundamental  Equations  of  Dynamics 

promising  denial  that  the  goal  can  be  aught  else  than  empirical 
rules,  ingenuity  being  restricted  to  embodying  best  in  them  the 
ascertained  data;  or  secondly,  in  questioning  doubt  how  the 
boundary-line  runs  among  special  cases.  Troubles  of  the  latter 
origin  involve  no  radical  divergencies,  since  they  are  everywhere 
inherent  in  such  a  separation  of  two  classes,  both  being  acknowl- 
edged to  exist.  Positions  like  the  first  mentioned  would  be  a 
fetter  upon  growth  through  their  exclusive  blindness  to  patent 
and  historic  facts,  were  not  a  saving  clause  inserted  in  extremist 
tenets  by  human  readiness  to  lapse  into  inconsistency  for  good 
cause.  To  illustrate  how  the  main  contention  spoken  of  would 
cramp  effort,  we  find  place  for  a  quotation,  which  however  is 
content  to  set  two  standards  in  opposition:  "Die  Fourier'sche 
Theorie  der  Warmeleitung  kann  als  eine  Mustertheorie  bezeichnet 
werden.  Dieselbe  .  .  .  griindet  sich  auf  eine  beobachtbare 
Tatsache  nach  welcher  die  Ausgleichungsgeschwindigkeiten 
[kleiner]  Temperaturdifferenzen  diesen  Differenzen  selbst  pro- 
portional sind.  Eine  solche  Tatsache  kann  zwar  durch  feinere 
Beobachtungen  genauer  festgestellt  werden,  sie  kann  aber  mit 
andern  Tatsachen  nicht  in  Widerspruch  treten.  ,  .  .  Wahrend 
eine  Hypothese  wie  jene  der  kinetischen  Gastheorie  .  .  .  jeden 
Augenblick  des  Widerspruchs  gewartig  sein  muss"  (Mach, 
Prinzipien  der  Warme,  page  115).  We  know  that  the  goal  here 
implied  for  theory  is  only  the  starting-post  for  it  in  the  doctrine 
of  another  school  of  thinking;  but  must  abstain  from  even 
outlining  the  argument. 

The  important  concern  for  dynamics  here  turns  plainly  upon 
the  question  of  aligning  it  in  method  with  the  rest  of  mathematical 
physics,  or  of  excepting  it  from  partnership  in  a  search  for  con- 
fessedly empirical  rules.  In  point  of  fact,  this  one  undeniably 
fruitful  wielding  of  idealized  conditions  has  been  a  bulwark  of 
defense  for  universal  procedure.  No  interested  student  can 
afford  to  neglect  Poincar^'s  pronounced  judgment  in  this  field, 


Notes  to  Chapters  I-IV  207 

to  be  found  especially  under  the  four  book-titles:  La  science  et 
riiypothese;  La  valeur  de  la  Science;  Science  et  M^thode; 
Dernieres  Pens^es.  The  first  three  are  most  compactly  accessible 
in  one  volume  of  English  translation  headed  The  Foundations  of 
Science  (1913);  the  fourth  not  included  in  that  collection  is  of 
recent  date  (1913)  and  presents  much  that  is  of  value.  Far  from 
putting  these  matters  aside  as  completed,  latest  developments 
have  renewed  and  intensified  their  Hvely  discussion.  As  repre- 
sentative in  one  direction  we  name  the  work  of  Robb :  A  Theory 
of  Time  and  Space  (1914);  and  on  another  line  a  paper  by  N. 
Campbell  (1910),  The  Principles  of  Dynamics  (Philosophical 
Magazine,  XIX,  page  168).  These  will  sufiiciently  lay  out  a 
track  for  further  pursuit,  in  connection  with  notes  1,  4  and  6. 

Note  6  (page  24).  There  is  much  more  here  than  the  kine- 
matical  colorlessness  that  precedes  the  introduction  of  dynamical 
elements.  Attention  is  being  directed  to  that  stage  of  inclusive 
preparedness  in  the  fundamental  equations  that  is  one  permanent 
attribute  of  "Analytic  mechanics,"  in  so  far  as  its  forms  of 
statement  are  made  equally  ready  to  contain  various  speciaUzed 
data.  Workers  in  the  subject  really  avail  themselves  of  this 
privilege  to  delay  in  particularizing.  Lorentz  for  example  does 
not  attempt  to  settle  in  advance  which  reference-frames  meet  the 
conditions  attached  to  the  primary  relations  for  the  electro- 
magnetic field.  He  lays  the  decision  aside  temporarily  with  the 
passing  remark  that  the  equations  remain  vahd  so  long  as  they 
accord  with  the  value  (c  =  3  X  10^°  cm./sec.)  for  light-speed  in 
free  space.  So  a  top's  local  behavior  relatively  to  the  earth's 
surface  follows  equations  of  motion  in  common  with  the  gyro- 
scopic compass  up  to  a  certain  divergence-point,  though  the 
former  ignores  the  earth's  rotation,  and  the  latter  may  be  said 
to  reveal  it.  In  a  group  of  parallel  cases  the  differences  center 
upon  replacing  gravitation  by  weight;  which  illustrates  how 
essentially  the  standards  of  desirable  or  attainable  precision 
enter  into  adapting  broader  analytic  expressions. 


208  Fundamental  Equations  of  Dynamics 

Note  7  (page  26).  A  number  of  points  touching  the  fuller  in- 
corporation of  vectors  into  physical  purposes  must  become  more 
definite  presently,  as  the  novelty  of  their  use  subsides.  Con- 
ventions that  have  been  transferred  from  mathematical  defini- 
tions, or  that  have  been  added  tacitly,  will  be  opened  to  needed 
revisions  first  by  being  made  explicit.  The  text  will  be  found  to 
adopt  this  feature  of  sound  policy  at  several  places,  none  of  which 
should  be  slurred.  Care  to  delimit  equivalences  legitimately  in 
relation  to  physical  conclusions  is  one  leading  idea  as  regards 
substitutions  that  approves  itself  to  be  a  needed  refinement 
upon  the  looser  term  equality.  For  accelerating  the  center  of 
mass  of  a  system  forces  have  the  quality  of  free  vectors,  because 
their  position  is  without  effect  upon  equivalence  in  this  respect. 
Yet  when  we  discuss  motion  relative  to  the  center  of  mass, 
forces  fall  away  from  that  equivalence,  being  then  dependent 
upon  position  for  their  effect,  and  consistently  they  cease  to  be 
free  vectors.  Such  instances  compel  us  to  qualify  classifications 
and  permissible  substitutions. 

Similar  deliberateness  in  borrowing  from  mathematics  is  en- 
couraged in  section  68,  with  its  suggested  distinction  between 
triangle  and  parallelogram  as  graphs  of  a  vector  sum;  and  in 
section  74,  where  an  element  of  parallel  shift  enters  to  round  out 
the  variableness  of  a  vector  quantity. 

The  idea  of  vector-angle  used  in  equation  (2)  has  not  yet 
found  its  way  into  textbooks.  Its  introduction  is  an  almost 
self-evident  detail  of  any  systematic  vector  algebra,  to  supply 
the  missing  member  of  the  series  in  which  angular  velocity  and 
acceleration  were  long  since  recognized.  How  that  proves  help- 
ful is  elaborated  in  section  92  and  its  sequel.  The  simple  step 
of  completing  with  natural  orienting  unit-vectors  the  established 
ratio  (ds/r)  for  magnitude  of  angle  seems  to  be  announced  first 
in  the  Physical  Review  (N.  S.),  I,  page  56  (1913).  In  section 
46  the  text  opens  from  this  side  a  new  meaning  for  the  rotation- 


Notes  to  Chapters  I-IV  209 

vector  that  fits  usefully  in  several  ways,  though  it  is,  of  course, 
nothing  but  that  second  interpretation  possible  for  every  vector 
product  which  happens  to  have  been  overlooked  here.  We 
must  ascribe  the  oversight  to  a  continuance  of  the  earlier  exclu- 
sive habit  of  using  only  the  projection  of  (r)  that  is  perpendicular 
to  (w),  and  not  the  corresponding  projection  of  the  latter  vector. 
Notice  how  the  rotation-vector  can  be  given  another  role  if  we 
rewrite  equation  (44)  in  the  form 

V  =  -  (r  X  <o), 

reading  the  second  member  as  the  negative  moment  of  (w) 
distributed  locally  at  each  (dm) .  This  has  important  connections 
with  the  uses  of  vector  potential,  and  the  association  of  the  curl 
operator  with  the  latter. 

Note  8  (page  31).  Later  research  has  come  to  the  aid  of 
mathematical  demands  or  convenience  on  this  side,  bj'  detecting 
real  transitions  with  however  sharp  gradient  behind  most  first 
'assumptions  of  discontinuous  break.  In  proportion  as  facts  of 
that  character  gather  they  soften  the  impression  of  artifice  in 
making  phenomena  amenable  to  treatment  by  allowing  for  quick 
gradations,  and  inchne  modern  physics  away  from  recognizing 
discontinuous  change  except  upon  compulsion.  See  Lorentz, 
The  Theory  of  Electrons  (1909),  page  11.  This  accounts  prob- 
ably for  some  psychology  alongside  the  mathematical  needs 
mentioned  in  section  26,  of  which  we  might  admit  an  admixture 
in  the  satisfaction,  when  identity  preserved  or  at  least  quantitj^ 
conserved  is  attributable  anywhere  without  too  strained  devices. 
Poincare's  shrewd  remark  is  to  this  effect:  "Physicists  can  be 
relied  upon  to  find  something  else  whose  total  remains  invariant, 
should  energy  leave  them  in  the  lurch."  And  is  there  not  some 
shade  of  disappointment  in  conceding  our  failure  to  trace  indi- 
vidual elements  of  energy  by  Poynting's  theorem,  as  well  as  the 
paths  of  flux?     Compare  Lorentz,   The  Theory  of  Electrons, 


210  Fundamental  Equations  of  Dynamics 

page  25;  Heaviside,  Electromagnetic  Theory,  I,  page  75 
(1893). 

Note  9  (page  33).  To  follow  lines  that  are  accommodated  to 
some  directive  idea  of  constancy  gives  in  many  ways  a  natural 
order.  About  this  we  should  acknowledge  though,  how  inevi- 
tably our  assigning  conceptually  common  or  constant  values  takes 
its  suggestion  from  what  are  means  or  averages  in  their  experi- 
mental basis.  Neither  must  the  truth  be  forgotten  with  which 
section  69  closes.  The  enlargement  in  appUcation  through  free 
use  of  mass-averages,  time-means,  and  the  like  can  be  instanced 
for  the  immediate  connection  from  sections  20,  21  and  31.  But 
it  confronts  us  without  any  special  search  everywhere  in  physics, 
when  we  remember  that  the  point  at  which  values  are  admitted 
to  be  "local"  is  in  practice  solely  a  matter  of  scale;  they  are 
finally  representative  of  mean  values  to  a  certain  order  of 
precision  (compare  section  42).  Less  familiar  but  perhaps  just 
as  significant  is  that  reading  of  the  curl  and  the  divergence  locally 
in  a  vector  field  which  sees  in  them  the  specification  of  an  arti- 
ficial symmetry  which  rests  upon  mean  values,  and  replaces 
legitimately  for  certain  ends  the  actual  field-distribution.  See 
the  Physical  Review,  XXXIV,  page  359  (1912);  Boussinesq, 
Note  sur  le  potentiel  spherique,  pages  319-329,  in  his  Application 
des  Potentiels  a  Tetude  de  FEquilibre  et  du  Mouvement  des 
Solides  elastiques  (1885). 

Note  10  (page  36).  Every  such  element  that  is  force-moment 
presents  a  local  resultant,  similar  to  those  met  in  section  19 
through  being  normal  to  the  individual  plane  of  its  factors.  As 
vector  products  these  local  resultants  are  all  open  to  the  same 
sort  of  double  reading  as  is  brought  up  for  the  rotation-vector 
in  note  7  and  completed  in  note  16.  The  process  of  mass- 
summation  for  a  system  then  continues  associated  with  the 
resultant  elements  (dH)  or  (dM),  combining  each  group  as  a 
vector  sum  to  a  total  resultant  of  determinate  orientation  and 


Notes  to  Chapters  I-IV  211 

tensor.  The  fraction  of  this  last  resultant  effective  or  available 
in  relation  to  any  particular  axis  of  unit-vector  (ai)  is  ascertain- 
able by  one  final  projection,  representable  respectively  bj'' 

H(a)  =  ai(H-ai);  M(a)  =  ai(M-ai). 

The  departure  from  the  cartesian  scheme  consists  especially  in 
reserving  projection  for  the  closing  operation,  to  be  executed  only 
when  the  demand  for  it  enters.  There  is  the  common  inversion 
of  order  between  mass-summation  and  projection,  on  passing 
over  to  vector  algebra. 

Note  11  (page  43).  There  is  a  considerable  region  opened  to 
plain  sailing  among  developments  hke  those  of  sections  32-35, 
whenever  the  observed  material  justifies  our  major  premiss  that 
inertia  occurs  as  a  variable  quantity.  But  whatever  general 
bearings  may  be  obtained  thus,  we  do  not  of  necessitj^  touch 
the  source  of  the  inferred  variableness,  and  much  less  do  we  reach 
a  halting-place  about  it  in  default  of  supplementary  evidence. 
The  emphasis  of  the  text  is  focussed  upon  the  truth  of  this  remark 
which  is  of  wide  application,  the  electronic  case  being  included 
among  others.  Consequently  there  is  a  warning  implied  to  avoid 
a  pitfall:  ascribing  prematurely  the  appearing  variableness  to 
one  tj^pe  of  source  among  several  of  which  experience  has  made 
us  aware,  and  thereby  affecting  the  conclusions  with  fallacy. 
The  conscious  fictions  that  cluster  round  the  idea  of  effective 
mass  should  make  us  wary  of  deceptive  illusions  there  whose 
enigma  has  not  been  resolved.  The  capacity  of  an  unincluded 
(or  undetected)  force  to  compel  indirect  recognition  of  itself 
in  the  inertia-coefficient  is  well-known.  And  a  long  line  of 
suggestive  connections  with  processes  of  continuously  repeated 
impact  have  their  root  in  an  old  problem.  This  is  the  trans- 
mission of  elastic  deformation  through  a  bar  struck  at  one-  end 
(see  Clebsch,  Theorie  d'Elasticite  des  corps  soUdes,  translated 
by.  St.  Venant,  page  480a,  Note  finale  du  §  60).     A  po.ssible 


212  Fundamental  Equations  of  Dynamics 

modification  of  that  treatment  for  impact  has  been  set  forth 
repeatedly,  in  the  attempt  to  cover  wider  conditions  of  converting 
and  storing  energy  within  a  system,  under  some  form  of  structm-e 
or  arrangement.  Heaviside  especially  has  achieved  instructive 
results  under  that  heading.  The  cogency  of  the  logic  in  trans- 
ferring demonstrated  consequences  of  this  nature  to  electrons 
hinges  on  the  query  in  how  far  the  convective  energy  of  electro- 
magnetic inertia  is  adequately  analogous  to  the  kinetic  energy  of 
(ponderable)  mass.  At  this  date  it  would  plainly  beg  the  larger 
question  to  assert  unreservedly  that  both  these  forms  of  energy 
are  literally  the  same. 

Note  12  (page  44).  In  the  closing  chapter  of  his  Kritische 
Geschichte  der  allgemeinen  Prinzipien  der  Mechanik  (1877) 
Diihring  urges  the  sound  advice  not  to  stop  short  of  first-hand 
contact  with  the  notable  contributions  that  mark  epochs  of 
advance.  The  case  of  d'Alembert's  discovery  enforces  the 
wisdom  of  that  counsel,  because  a  tradition  echoing  an  imperfect 
apprehension  of  the  principle  has  leaned  toward  perverting  the 
gist  of  it  from  the  meaning  that  the  leaders  in  dynamics  state 
clearly,  whose  essential  thought  sections  38-41  aim  to  restore. 
Compare  them  with  the  analysis  of  the  principle  in  Mach's 
Science  of  Mechanics  and  in  Helm's  Energetik.  One  source  of 
confusion  can  be  located  in  the  transposition  that  yields  the 
forms  of  equation  (38).  This  point  is  alluded  to  at  the  end  of 
section  41;  and  the  idea  is  expanded  with  elementary  detail  in 
Science,  XXVIII,  page  154.  Some  obstacles  to  ready  under- 
standing are  due  no  doubt  to  a  certain  crabbed  brevity  of  the 
nascent  formulation  in  d'Alembert's  Traite  de  Dynamique 
(1758),  found  in  Chapter  I  of  Part  II.  A  German  translation  of 
this  classic  is  provided  among  Ostwald's  Klassiker  der  exakten 
Wissenschaften  (Number  106). 

Note  13  (page  51).  The  influence  of  the  energetic  view  per- 
vades the  handling  of  energy  flux  and  of  the  accompanying  forces 


Notes  to  Chapters  I-IV  213 

or  stresses.  The  transfer-forces  of  the  text  appear  for  example 
in  Helm's  exposition  (Energetik,  page  233  and  passim).  The 
habit  of  thinking  in  these  terms  is  cultivated  by  greater  familiar- 
ity with  storage  of  energy  in  media,  which  has  added  the  vigor 
of  a  physically  conceived  process  to  the  formal  nature  of  potential 
energy  in  the  earliest  instance  of  gravitation,  where  the  mecha- 
nism remains  completely  obscure  (see  section  3).  It  is  growing 
increasingly  evident  how  the  outcome  of  explorations  among 
energies  intrinsic  and  external  is  capable  of  reduction  in  parallel 
fashion,  exhibiting  the  conditioned  modes  of  revealing  their 
presence  and  the  measured  extent  of  their  availability.  The 
lessons  about  cautious  inference  of  which  some  scant  mention 
is  made  in  the  text  are  perhaps  nowhere  more  impressive  among 
the  inductions  of  physics,  when  once  the  safety  of  non-committal 
attitude  must  be  abandoned  in  active  search  for  a  determinate 
process.  We  remember  the  remark  that  "An  infinite  number 
of  mechanical  explanations  are  possible"  (Poincare),  especially 
since  we  deal  primarily  with  finite  or  statistical  resultants; 
and  even  plausible  schemes  are  numerous  enough  to  leave  a 
broad  margin  for  indecision.  See  Lorentz,  Theory  of  Electrons, 
pages  30-32. 

Poynting's  original  paper  should  not  be  left  unread  (Philo- 
sophical Transactions  (1884),  Part  II,  page  343);  nor  touch  be 
lost  with  Heaviside's  stimulating  directness  (e.  g.,  Electromag- 
netic Theory,  I,  pages  72-78).  A  sensible  summary  incorporat- 
ing links  with  relativity  is  furnished  by  MattioH;  Nuovo  Cimento 
(series  6),  IX,  pages  255,  263  (1915). 

Note  14,  page  53.  Geometrical  conditions  are  always  a  need- 
ful auxiliary  in  expressing  constraints  for  the  reason  named  in 
the  text.  The  use  of  indeterminate  multipliers  would  carry 
unreduced  geometrical  forms  into  the  equations  of  motion,  giving 
what  might  be  called  quasi-forces.  Lagrange  himself  offers  that 
analysis  of  their  significance  in  his  Mecanique  Analytique,  I, 


214  Fundamental  Equations  of  Dynamics 

pages  69-73  (Bertrand's  edition  (1853)).  Later  practice  runs 
more  nearly  in  the  line  of  separating  these  supplementary  rela- 
tions from  the  purely  dynamical  truths,  and  using  the  former 
admittedly  as  mathematical  aid  in  eUminations  looking  to  ends 
like  integrations.  Thomson  and  Tait  held  it  part  of  their 
service  to  have  brought  together  the  fully  dynamical  treatment 
of  constrained  and  of  free  systems  (Natural  Philosophy,  Part  I, 
pages  271,  302). 

Note  15  (page  54).  The  point  now  reached  offers  occasion  to 
add  explicit  reference  to  Routh's  encyclopedic  work  in  two 
parts:  Elementary  Rigid  Dynamics,  Advanced  Rigid  Dynamics; 
as  a  storehouse  to  which  we  shall  long  resort  for  authoritative 
presentation  of  characteristic  material  in  this  field.  The  design 
of  our  text  has  acknowledged  as  one  main  object  to  foster  the 
study  of  masters  such  as  Kelvin,  Routh  and  a  few  others  in 
dynamics.  To  this  end  we  are  building  a  less  steep  approach  to 
the  level  upon  which  their  progress  moves.  It  cannot  be  said 
to  stand  in  prospect  that  these  writers  will  become  antiquated; 
but  need  will  arise  from  time  to  time  for  seeing  the  older  system- 
atic grouping  in  an  altered  perspective,  in  order  to  renew  connec- 
tions or  symmetry  that  temporary  stress  upon  some  lines  of 
growth  may  have  disturbed. 

Note  16  (page  57).  Preparation  has  been  made  by  anticipa- 
tion in  the  connection  of  notes  7  and  10  to  accept  this  meaning 
and  ojffice  for  the  rotation-vector  which  are  an  enlargement  upon 
the  usual  current  statement  about  it.  That  aspect  is  adapted  to 
set  in  higher  relief  its  comprehensive  and  yet  particular  relation 
to  those  individual  radius-vectors  upon  which  vector  algebra 
turns  attention.  There  is  some  advantage  gained,  too,  by 
approaching  the  special  rigid  connection  on  the  line  that  starts 
with  the  complete  freedom  in  equation  (2),  and  sees  the  vector 
(w)  of  common  apphcation  to  all  radius-vectors  to  be  an  out- 
growth of  that  rigidity. 


Notes  to  Chapters  I-IV  215 

Note  17  (page  64).  It  is  important  to  keep  track  of  successive 
restrictions  that  enter  to  affect  the  range  of  conclusions.  Here 
we  must  not  overlook  that  the  added  condition  of  rigidity 
influences  only  a  general  reduction  in  form  for  certain  parts  of 
(E,  H,  P,  M)  that  are  seen  to  occur  already  in  equations  (10,  12, 
54,  55)  as  written  for  any  non-rigid  system  of  constant  mass. 
In  brief,  the  notion  of  a  constituent  translation  with  the  center 
of  mass  applies  to  all  such  systems ;  and  so  does  the  independent 
treatment  of  that  translation  and  of  the  motion  relative  to  the 
center  of  mass,  as  spoken  of  in  section  52.  That  point  is  elab- 
orated for  elementary  purposes  in  my  Principles  of  Mechanics, 
Part  I,  pages  91-101.  Including  now  equations  (19,  20)  it  is 
made  fully  evident  how  no  new  situation  is  introduced  when  we 
ascribe  rigidity  to  the  body,  except  in  the  entrance  of  rotation. 
While  absorbing  the  residual  (E,  H,  P,  M),  this  type  of  motion 
also  gives  concise  expression  to  their  values,  in  every  one  of 
which,  it  will  be  noticed,  either  (o)  or  (w)  appears,  marking  the 
relation  of  both  to  the  body  as  a  whole. 

Note  18  (page  70).  The  frequent  necessity  of  a  dynamically 
active  couple  for  an  adjusted  control  securing  kinematical  con- 
stancy in  the  vector  (w)  is  now  an  everyday  lesson  learned  from 
the  directive  couple  of  rotation  about  a  fixed  axis.  The  possible 
divergence  of  (co)  and  (H)  furnishes  the  simple  key  which  cuts 
off  vector  constancy  of  both  together;  with  habitual  demand 
then  prevailing  for  some  (M)  associated  with  every  change  in 
(H).  But  there  has  been  an  astonishing  record  of  tenacious 
refusal  to  distinguish  between  such  conditions  of  active  control 
and  the  conditions  of  equilibrium,  here  and  in  the  companion 
instance  of  radial  control  requisite  for  continuance  of  circular 
motion.  The  surviving  power  of  instinctive  prepossessions  has 
perpetuated  in  unexpected  quarters  the  ancient  unclearness 
lurking  behind  "  centrifugal  force  and  couple  " ;  and  this  threatens 
to  endure  under  the  full  illumination  of  the  vector  view.  The 
15 


216  Fundamental  Equations  of  Dynamics 

root  of  many  like  confusions  is  traceable  to  a  failure  really  to 
grasp  the  facts  in  the  first  of  equations  (38) ,  with  unfaltering  dis- 
crimination between  impressed  and  effective  forces.  That  equa- 
tion does  not  describe  an  actual  equilibrium;  neither  does  the 
result  of  any  transposition  which  yields  an  equation  like  the 
second  form  of  (77).  Yet  compare  the  presentation  by  authori- 
ties: Klein  and  Sommerfeld,  Theorie  des  Kreisels  (1897),  pages 
141,  166,  175,  182;  though  no  criticism  applies  anywhere  to  their 
mathematical  correctness. 

Note  19  (page  82).  This  labored  insistence  upon  the  dual 
aspects  of  all  coincidences  is  indeed  designed  to  remove  an 
ambiguity  in  symbolism  whose  currency  has  grown  out  of  im- 
perfect attention  to  them.  There  is  usually  reward  for  watch- 
fulness on  those  points.  But  the  allowableness  of  such  detail 
in  the  text  rests  more  upon  its  initiative  for  developing  the  idea 
of  shift  in  section  79.  Notice,  as  we  proceed,  how  often  the 
unit-vectors  and  the  tensors  of  vector  quantities  offer  themselves 
naturally  as  independent  variable  elements,  and  afford  a  ground 
for  partial  differentiations  of  a  type  peculiar  to  vector  algebra. 

Note  20  (page  88).  Of  course  forces  are  "bound  to  super- 
position" only  by  the  same  tie  of  definition  or  specification 
that  holds  velocity  and  acceleration  also,  and  that  is  broken 
when  we  abandon  the  parallelogram  graph.  But  it  is  remark- 
able how  regularly  in  physics  that  mutual  independence  among 
energies  (and  among  forces  that  change  them)  is  experimentally 
supported,  of  which  superposition  and  linear  relation  are  mathe- 
matical expression.  Still  it  is  reasonable  to  grant  that  not  all 
definitions  devised  for  physical  quantity  have  escaped  a  bias 
from  this  side  which  will  need  to  be  allowed  for  or  rectified. 
Yet  the  high  price  paid  for  relinquishing  that  simplest  rule 
warrants  the  change  of  base  only  on  clearest  showing  of  the 
balance-sheet. 

By  referring  to  "physical  status"  the  text  means  to  encourage 


Notes  to  Chapters  I-IV  217 

that  scrutiny  for  terms  of  algebraic  origin  whose  favorable  and 
unfavorable  outcome  in  particular  connections  it  cites  in  several 
places.  To  be  sure,  candor  and  detachment  are  called  for  con- 
tinually in  reaching  judgment  through  the  arguments  by  con- 
vergent plausibility  upon  which  closing  of  the  doubtful  issues 
here  depends  (see  sections  6  and  7). 

Note  21  (page  93).  The  superficial  features  of  what  is  here 
named  shift  are  detectable  generally  in  previous  accounts  of 
coordinate  systems;  and  Hay  ward  is  often  credited  with  a  com- 
prehensive survey  of  the  subject  in  a  paper:  On  a  direct  method 
of  estimating  velocities  with  respect  to  axes  movable  in  space 
(Cambridge  Philosophical  Transactions  (1864),  X,  page  1*). 
Anticipations  of  the  controlling  purpose  in  shift  might  be  ex- 
pected confidently,  since  its  ramifications  are  now  recognizable 
through  all  that  coordinate  machinery  of  early  devising  without 
which  commonest  operations  of  algebra  would  have  been  blocked. 
But  the  circumstance  seems  exceptional  that  completed  analysis 
of  its  working  has  been  postponed.  The  proposition  presented 
by  equation  (137)  does  not  occur  in  the  first  editions  of  Routh, 
and  he  never  gives  to  it  deserved  prominence.  Abraham's  state- 
ment of  it  is  of  course  formally  right,  yet  he  describes  our 
(X'Y'Z')  questionably  as  a  "Rotierendes  Bezugssystem "  (The- 
orie  der  Elektrizitat  (1904),,!,  page  34).  The  relations  of 
coincidence  that  make  equation  (124)  important  Routh  disposes 
of  in  one  obscurely  placed  line:  "As  if  the  axes  were  fixed  in 
space"  (Elementary  Rigid  Dynamics  (1905),  page  213).  Equally 
casual  is  Abraham  (p.  115) :  "Die  Umrechnung  [auf  ein  bewegtes 
Bezugssj^stem]  geschieht  genau  so,  als  ob  das  bewegte  System  in 
seiner  augenblicklichen  Lage  ruhte."  This  comparative  blank 
left  place  for  that  more  systematic  or  conscious  display  which 
vector   algebra   favors   of   the   really   operative   methods.     Its 

*Thb  is  the  date  of  publication.     The  paper  itself  was  dated  and  read 

(1856). 


218  .    Fundamental  Equations  of  Dynamics 

partial  novelty  has  set  its  measure  at  a  length  in  the  text  that 
may  well  be  curtailed  when  their  leading  thought  has  once  been 
laid  down. 

Note  22  (page  98).  Some  authors  cover  the  point  by  a  dis- 
tinction between  explicit  and  implicit  functions  of  time.  Or 
again  the  changing  relation  fairly  equivalent  to  our  shift  of 
(i'j'k')  among  (ijk)  is  made  to  introduce  a  partial  time-derivative 
(Thomson  and  Tait,  Natural  Philosophy,  Part  I,  page  303). 
It  cannot  escape  notice  what  direct  gain  in  clearness  the  regular 
acceptance  in  our  algebra  of  time-derivatives  for  unit-vectors 
yields.  The  due  adjustment  of  pace  for  shift,  especially  in 
order  to  simplify  dynamical  problems  in  astronomy,  has  called 
forth  important  discussion  touching  the  double  entry  of  time, 
while  methods  of  treating  perturbations  were  becoming  fully 
established;  and  this  engaged  the  attention  of  men  like  Donkin, 
Jacobi,  Hansen.  There  is  a  sequel  in  that  region  to  sections 
107-112;  see,  for  instance,  Cayley,  Progress  in  Theoretical 
Dynamics,  British  Association  Report  (1857). 

Note  23  (page  109).     The  type  to  be  remarked  in  equations 

(154)  as  leading  to  generalizations  of  them  is  the  functional 
relation  between  each  of  (x',  y',  z')  and  all  of  both  (x,  y,  z)  and 
(x,  y,  z).  The  same  combinations  show  reciprocally  when  equa- 
tions (150)  are  differentiated,  and  they  affect  characteristically 
the  expressions  derived  for  kinetic  energy.     In  equations  like 

(155)  the  first  equality  of  partial  derivatives  brings  out  the 
extent  to  which  building  up  is  occurring  in  the  instantaneous 
lines  of  (x',  y',  z');  and  the  second  such  equality  connects 
the  remainder  of  the  increment  visibly  with  changes  of  slope  that 
are  proceeding.  It  becomes  then  a  simple  matter  to  forecast 
how  these  constituents  will  reproduce  the  result  given  through 
a  vector  derivative. 

Note  24  (page  118).  One  main  objective  being  to  specify 
configurations  in  the  standard  frame,  it  is  indispensable  in  the 


Notes  to  Chapters  I-IV  219 

plan  that  some  unbroken  link  with  the  latter  should  be  main- 
tained. The  permanent  orientation  in  (Z)  of  the  angle-vector 
(tjf)  serves  that  purpose,  every  displacement  (dtjf)  being  im- 
mediately relative  to  (XYZ).  By  the  terms  of  section  93  dis- 
placements in  (•&)  have  this  one  step  interposed  between  direct 
junction  with  (XYZ);  and  finally  displacements  in  (^)  are  two 
removes  from  that  immediate  relation.  Taking  other  comment 
from  the  text,  it  is  made  apparent  how  adequately  all  this 
parallels  the  conception  of  displacements  parallel  to  (X,  Y,  Z) 
as  successive,  independent,  and  cumulatively  relative.  There 
too,  whichever  the  second  and  third  displacements  are,  according 
to  the  order  selected,  each  must  accept  a  determined  initial  state 
due  to  the  displacements  that  have  preceded  it.  The  residual 
difference  is  inherent  in  the  mutually  supplementary  qualities 
of  linear  and  angular  displacements.  Other  parallel  features 
with  longer-established  vector  schemes  will  repay  attention; 
for  example  the  sentence  just  preceding  equation  (174)  does  not 
mark  an  exceptional  condition.  It  is  of  interest,  too,  to  dwell 
upon  the  fact  implied  on  page  120,  that  (t|r,  ^,  ^)  give  us  the 
model  of  a  coordinate-set  with  a  changing  obliquity  among  its 
unit-vectors.  It  is  obviously  unessential,  except  for  conven- 
ience, that  (i'j'k')  should  be  orthogonal  or  retain  any  constant 
relative  obliquity.  Some  proposals  have  been  made  to  include 
the  more  general  relation  of  direction  for  sets  of  unit- vectors ; 
and  the  necessary  modification  of  section  45  would  be  no  more 
than  simple  routine. 

Note  25  (page  125).  Needless  to  say,  the  revised  conclusion 
reached  through  equation  (186)  renounces  any  attempt  to  make 
complete  derivatives  out  of  what  are  actually  partials;  but  it 
succeeds  in  assigning  their  proper  quality  to  derivatives,  for  all 
such  combinations  involving  vectors,  under  a  general  rule  stated 
at  the  close  of  section  100.  The  root  of  the  matter  goes  back  to 
equation  (124);    and  the  establishment  of  angle  among  vectors. 


220  Fundamental  Equations  of  Dynamics 

places  it  in  a  category  with  them  in  this  respect  also.  In  what 
form  the  omission  of  that  element  raises  the  diflSculty  may  be 
gathered  from  Klein  and  Sommerfeld,  Theorie  des  Kreisels, 
page  46.  The  truth  is  that  a  similar  non-integrability  of  tensor 
accompanies  every  plan  of  shift,  except  those  in  which  a  special 
condition  is  satisfied  that  includes  them  among  what  may  be 
classed  with  envelope  solutions  (see  section  116). 

Note  26  (page  137).  The  text  bears  frequent  testimony  con- 
sistently to  a  high  appreciation  for  the  genius  and  inspiration 
of  the  earlier  workers  who  built  dynamics,  among  whom  we  may 
name  Coriolis.  Yet  we  should  respect  our  obligation  also  to 
carry  forward  or  to  rectify  the  first  suggestions;  being  taught  to 
expect  advances  in  our  reading  attached  to  results  especially, 
whose  mathematical  accuracy  has  never  been  questioned.  It  is 
that  hint  of  possible  improvement  which  the  text  here  submits, 
afl&rming  the  lesson  of  cultivating  perception  of  physical  mean- 
ings upon  which  best  modern  thought  concentrates,  and  which 
is  illustrated  by  sections  35,  57  and  104;  all  to  be  taken  in 
the  light  of  repeated  comment  upon  those  clouding  transfers 
between  the  two  members  of  equation  (37)  which  are  still  too 
prevalent. 

Note  27  (page  141).  Hansen,  Sachsische  Gesellschaft  der 
Wissenschaften,  Mathematisch-physikalische  Klasse,  III,  pages 
67-71.  This  original  statement  retains  value,  partly  still 
through  the  material  it  discusses,  and  again  through  the  moral 
it  conveys  that  vector  methods  have  made  these  problems  more 
manageable.  The  reaction  of  Jacobi  in  some  letters  to  Hansen 
(Crelle,  Journal  fiir  reine  und  angewandte  Mathematik,  XLII, 
(1851))  shows  instructively  the  struggle  toward  clear  and  firmly 
grasped  thought  proceeding,  with  strictest  scrutiny  of  detail 
in  the  new  proposal.  In  the  paper  referred  to  above,  Hansen's 
double  use  of  time  is  worked  out  (compare  note  22),  that  remains 
current  among  astronomers. 


Notes  to  Chapters  I-IV  221 

Note  28  (page  155).  We  do  not  measure  rightly  the  inherit- 
ance of  rigid  dynamics  from  Euler's  labors  without  conscious 
effort  to  reconstruct  the  void  that  they  filled  once  for  all.  Unless 
his  inventive  intuitions  had  here  been  favored  by  a  happy  chance, 
he  could  hardly  have  moulded  from  the  first  heat  so  many  of  the 
forms  that  seem  destined  to  hold  permanent  place.  We  can 
imagine  that  his  inspiration  caught  early  glimpses  of  the  relation 
that  equations  (72)  and  (258)  now  convey;  but  Euler  may  have 
been  content  to  seize  the  validity  of  equation  (257)  without 
proving  it,  as  Fourier  did  in  like  case.  Certain  it  is  that  the 
point  involved  in  that  equivalence  seemed  troublesome  enough 
to  be  made  the  object  of  various  special  proofs,  before  our 
general  equation  (137)  had  been  attained  (see  Routh,  Elementary 
Rigid  Dynamics  (1882),  page  212).  For  the  historic  date,  the 
memoir  presented  to  the  Berlin  Academy  is  quoted  (1758). 
But  a  satisfactory  survey  of  Euler's  contributions  on  the  topic 
is  best  obtained  through  his  collected  works.  Easier  access 
perhaps  is  had  in  the  German  translation  (Wolfers,  1853);  in 
the  volumes  3^  entitled  Theorie  der  Bewegung  the  "Centrifugal 
couple"  appears  at  page  323,  and  our  main  interest  would  prob- 
ably concentrate  on  pages  207-443. 

Note  29  (page  169).  Klein  and  Sommerfeld,  Uberdie  Theorie 
des  Kreisels  (1897-1910),  is  one  instance,  quoting  our  Preface, 
how  special  treatises  of  unquestioned  excellence  make  superfluous 
an  attempt  to  replace  them.  This  work,  and  Routh's  version  in 
the  Advanced  Rigid  Dynamics  (edition  of  1905),  Chapter  V, 
with  Thomson  and  Tait's  discussions  passim  in  Natural  Philos- 
ophy, Part  I,  supply  for  gyroscopic  problems  the  indispensable 
material,  exhaustive  of  more  than  their  general  aspects.  The 
aim  of  the  text  here  is  strictly  confined  to  lending  its  announced 
special  emphasis  to  two  items.  One  is  shown  to  be  of  ramifying 
importance  as  a  singular  value  round  which  deviations  from  it 
may  be  organized;    the  other  is  uniquely  characteristic,  and  it 


222  Fundamental  Equations  of  Dynamics 

proves  amenable  to  this  analysis  most  simply,  in  comparison 
with  other  methods.  Compare  in  verification  Theorie  des 
Kreisels,  pages  247,  316,  on  strong  and  weak  tops. 

Note  30  (page  180).  A  fuller  command  of  generalized  co- 
ordinates and  forces  as  an  effective  working  method  can  be 
inferred  from  evidence  on  two  sides:  first,  more  unequivocal 
recognition  is  accorded  to  their  finally  scalar  type;  and  secondly, 
the  primary  demonstration  of  relations  shows  increasingly 
directer  insight.  Dispose  of  the  latter  point  by  collating 
Lagrange's  proof  (Mecanique  Analytique,  I;  Dynamique,  Sec- 
tion IV);  Thomson  and  Tait,  whose  change  between  (1867) 
and  (1879)  is  instructive;  and  Heaviside,  Electromagnetic 
Theory,  III,  page  178.  The  last-named  is  a  climax  of  condensa- 
tion, and  thereby  somewhat  unfitted  for  the  text;  but  it  will 
be  quoted  below  for  a  double  reason.  The  quantitative  emanci- 
pation of  Lagrange's  equations  may  be  traced  gradually,  if  we 
like,  beginning  with  equations  such  as  (150,  151),  where  the 
(1,  m,  n)  coefficients  are  particular  reduction  factors  conditioned 
as  in  equation  (152).  Next  advance  to  the  more  liberal  possi- 
bilities of  linear  vector  functions  illustrated  by  equations  (86, 
89),  and  clinch  the  series  with  Byerly's  half-humorous  emphasis 
(Generalized  Coordinates  (1916),  page  33).  This  book  has  the 
merit  of  helpfully  discursive  approach  to  a  large  subject;  and 
though  it  seems  tacitly  limited  to  the  vector  conception,  closing 
the  matter  on  the  range  that  Lagrange  occupied  at  one  bound 
and  not  gradually,  proper  antidote  can  be  sought  elsewhere. 
See  Silberstein,  Vectorial  Mechanics  (1913),  page  59;  while 
Ebert  has  been  referred  to  in  note  3,  for  his  treatment  in  the 
larger  spirit  of  energetics. 

We  insert  now  the  quotation  from  Heaviside;  it  illustrates 
fairly  the  ne  plus  ultra  in  both  respects.  Notation  of  our  text 
is  continued.  Because  (E)  is  a  homogeneous  quadratic  function 
of  the  velocities,  Euler's  theorem  about  homogeneous  functions 
enables  us  to  .write 


Notes  to  Chapters  I-IV  223 

of  which  the  legitimate  total  time-derivative  is 

Since  (E)  is  ''by  structm-e"  a  function  of  velocities  and  co- 
ordinates only, 

Divide  the  last  equation  by  (dt)  and  subtract  from  the  second, 
giving 

the  last  member  expressing  the  energetic  in  variance  of  activity 
(see  equation  (298)). 

It  would  be  misleading  if  the  text  pretended  to  do  more  than 
give  Lagrange's  equations  their  setting  of  introductory  connec- 
tion with  the  other  topics  treated.  In  order  to  proceed  safely 
the  results  here  gleaned  must  be  followed  up  seriously;  the 
references  given  already  indicate  where  to  begin,  and  they  can 
be  relied  upon  to  supplement  themselves  as  the  subject  opens. 
Questions  to  be  met  at  once  are  alluded  to  incidentally  in  section 
136:  a  rationally  consistent  view  of  superfluous  coordinates, 
including  how  they  may  drop  that  character  and  become  physical; 
and  the  bearing  of  that  quoted  "interlocking"  upon  the  signifi- 
cance of  the  term  holonomous.  That  there  are  more  vital  issues 
awaiting  analysis  is  suggested  by  Burbury  (Proceedings  of  the 
Cambridge  Philosophical  Society,  VI,  page  329);  by  such  com- 
ment as  Heaviside's  (Electromagnetic  Theory,  III,  page  471) 
upon  Abraham's  successful  extension  of  Lagrange's  equations; 
and  by  the  lines  of  inquiry  to  which  note  32  points. 


224  Fundamental  Equations  of  Dynamics 

Note  31  (page  194).  This  development  is  seen  to  be  borrowed 
from  Thomson  and  Tait,  pages  320-24.  The  few  changes  are 
adapted  here  and  there  to  an  even  keener  intent  to  keep  the 
energies  and  momenta  at  the  front,  subordinating  the  investiture 
with  mathematics.  It  was  thought  needful  to  drive  the  entering 
wedge  before  closing,  for  the  sake  of  those  continuations  to  which 
Maxwell's  example  leads.  The  reduction  factors  (1,  m,  n)  are 
easily  released  from  their  trigonometrical  meaning,  and  other 
geometrical  implications  cancelled. 

Note  32  (page  200).  For  the  justified  application  of  equation 
(333),  or  of  forms  derivable  mathematically  from  it,  to  all  se- 
quences of  energy  change,  one  turning-point  is  set  by  delimiting 
the  necessary  equivalences  between  the  mechanical  readings  of 
(E)  and  (<l>)  and  the  broader  dynamical  ones.  This  general  idea 
is  pursued  by  Konigsberger  in  his  papers,  Uber  die  Prinzipien 
der  Mechanik  (Sitzungsberichte  der  Berliner  Akademie  (1896), 
pages  899;  1173);  and  is  entertained  by  Whittaker  in  his 
Analytic  Dynamics  (1904),  Chapter  X,  passim.  The  stimulus 
to  this  quest  seems  still  attached  to  the  possibility  of  construct- 
ing a  parallel  in  mechanical  energy  by  using  values  connected 
with  other  energy  changes.  One  gathers  this  meaning  from  the 
utterance  of  Larmor  (Aether  and  Matter,  page  83)  and  others 
like  it. 


INDEX 


The  Numbers  refer  to  Pages 


Abraham,  217,  223. 

Absolute  measure,  5. 

Absolute  motion,  9,  10. 

Acceleration,  83;  and  center  of  mass, 
37,  61;  and  ideal  coordinates,  144- 
147;  and  Newton's  second  law,  33; 
and  shift,  150,  152;  and  tangent- 
normal,  148,  angular,  62;  in  rota- 
tion, 65-6;  in  space  curves,  151-2; 
invariance  of,  83,  90;  mass-average 
of,  37;  polar  components  of,  134- 
136;  relative  to  center  of  mass,  61; 
transfer  for,  89;  uniplanar,  136, 
150. 

Activity,  35,  36,  201. 

Adjustments,  and  equilibrium,  69, 
174,  216;  and  force-moment,  69, 
174;  imaginary,  175,  177,  179;  of 
shift,  97. 

Angle,  and  moment  of  momentum, 
27. 

Angle-vector,  58,  208,  219. 

Angular  acceleration,  62;  and  force- 
moment,  67-8,  69.  71-2;  and  shift, 
126-131,  160-161,  164-165;  axis 
of,  66;  base-point  for,  66;  transfer 
for,  124. 

Angular  displacement,  27,  56,  58,  79, 
115. 

Angular  velocity,  57;  base-point  for, 
57;  transfer  for,  124. 

Approximation,  17,  18,  203;  and 
particle,  29;  and  rigid  dynamics, 
53. 


Atomic  energy,  51. 
Average  acceleration,  37. 
Axes,  principal,  73,  157,  162. 
Axis,  of  angular  acceleration,  66;  of 
rotation- vector,  58. 

Base-point,  for  angular  acceleration, 
66;  for  angular  velocity,  57;  for 
force-moment,  36;  for  moment  of 
momentum,  26-27. 

Bodies,  system  of,  16. 

Body,  16;  continuous  mass  of,  16; 
homogeneous,  31. 

Boussinesq,  210. 

Burbury,  223. 

Byerly,  222. 

Campbell,  207. 

Cartesian  coordinates,  and  funda- 
mental quantities,  113;  and  shift, 
107-111;  scalar  character  of,  112, 
204. 

Cayley,  218. 

Center  of  mass,  28;  acceleration 
relative  to,  61;  and  mean  accelera- 
tion, 37;  and  energy,  55,  60,  64; 
and  force-moment,  60,  63;  and 
impressed  "force,  48-49,  63;  and  in- 
variance, 83;  and  moment  of  mo- 
mentum, 55,  60,  83;  and  momen- 
tum, 29,  63;  and  particle,  37-38, 
63;  and  power,  60,  63;  and  pure 
rotation,  64-65,  75 ;  and  rigid  solids, 
55,  60-64;  and  total  force,  37,  48. 


225 


226 


Fundamental  EquaMons  of  Dynamics 


The  Numbers  refer  to  Pages 


63;  and  translation,  37-38,  63,  215; 
and  velocity,  29,  55,  57;  rotation 
about,  55.  57,  60-63;  velocity  rela- 
tive to,  55,  57. 

Centimeter-gram-second  system,  21. 

Centrifugal  couple,  68,  215. 

Centrifugal  force,  compound,  137, 
220. 

Clebsch,  211. 

Coincidence,  dual  nature  of,  82,  91, 
216. 

Comparison-frame,  78;  and  accelera- 
tion, 89;  and  shift,  94,  96,  97,  104; 
and  velocity,  82-88;  notation  for, 
78;  velocity  of,  85-88. 

Comparisons,  timeless.  81. 

Compoimd  centrifugal  force,  137, 
220. 

Concepts,  physical,  8,  19. 

Conditions,  geometrical,  52,  213. 

Configuration,  78-79,  181-182,  187. 

Configuration  angle,  79-80;  and  shift, 
123,  124,  126-131;  derivatives  of, 
117-123. 

Configuration  angles,  Euler's,  114, 
219;  and  rotation-vector,  117-123; 
partial  derivatives  of,  125,  219. 

Connections,  internal,  49-50,  55; 
transmit  force,  50. 

Conservation  of  energy,  4. 

Conservative  system,  5,  7. 

Constancy,  of  mass,  25;  simplifica- 
tion by.  33,  54. 

Constraints,  3,  47,  68,  140;  and  La- 
grange equations,  187;  and  pure  ro- 
tation, 68,  75;  and  rigid  solids,  53, 
55,  62. 

Continuity,  34,  209;  of  density,  31; 
of  mass,  16. 

Convection,  of  energy,  45;  of  momen- 
tum, 45. 


Conversions  of  energy,  45,  52. 

Coordinates,  and  configuration,  181- 
182,  187;  Euler's,  114;  generalized, 
179;  ideal,  141-147;  ignoration  of, 
194-200,  224;  oblique,  116,  219; 
polar,  130-135;  standard  frame, 
112;  superfluous,  183,  190,  191, 
223;  tangent-normal,  147;  and 
shift,  97. 

Coriolis,  137,  220. 

Couple,  36,  63,  68;  centrifugal,  68, 
215;  directive,  215. 

D'Alembert,  2.  7,  8,  50,  53,  212;  and 
equation  of  motion,  53;  and  La- 
grange, 180,  184;  and  Newton's 
third  law,  50. 

D'Alembert's  principle,  5C;  and  im- 
pulse, 50. 

Defining  equalities,  23,  44,  48. 

Definitions:  activity,  35;  angular 
acceleration,  62;  angular  velocity, 
57;  body,  16;  constraints,  47;  cen- 
ter of  mass,  28;  effective  force,  48; 
force,  34,  36-38;  force-moment, 
35,  36;  impressed  force,  48;  inertia, 
6;  kinetic  energy,  22,  25,  mean 
vector,  28;  moment  of  momentum, 
22,  27;  momentum,  22;  power,  35, 
36;  rotation- vector,  57;  system  of 
bodies,  16;  translation,  27. 

Degrees  of  freedom,  178,  182;  and 
equations  of  motion,  183. 

Density,  and  volume-integral,  30,  31; 
continuity  of,  31. 

Derivatives,  of  tensors,  93,  96,  102, 
154;  partial  and  total,  128,  218, 
219. 

Descriptive  vectors,  137,  141. 

Differentiation,  of  mass-summa- 
tions, 32-33. 


Index 


227 


The  Numbers  refer  to  Pages 


Direction-cosine  generalized,  190, 
222. 

Directive,  forces  and  power,  140: 
moment,  68,  215. 

Discover}',  of  principles,  18. 

Discrimination,  among  time-func- 
tions, 98,  218,  220. 

Displacement,  angular,  27,  56,  58, 
79,  115;  by  rotation,  56,  58. 

Distributed  vectors:  force,  34;  mo- 
mentum, 26;  transfer-forces,  45-46. 

Divergence:  angular  acceleration  and 
force-moment,  70,  74;  moment  of 
momentum  and  rotation  vector, 
67,  70,  215. 

Donkin,  218. 

Driving  pomt,  50. 

Diihring,  212. 

DjTiamical  equations,  Euler's,  155- 
166,  180,  192;  Lagrange's,  179-200. 

Dj-namical  systems,  16. 

Dj-namics,  and  kinematics,  9,  12,  13, 
72,  165-166;  and  Lagrange  equa 
tions,  7,  180;  and  mathematics,  1, 
36-37,  113,  137,  139,  174-175,  2C8, 
216,  220;  and  mechanics,  16;  fic- 
tions in,  8,  36-37;  of  precession, 
171-174;  stability  of,  2,  8,  15. 

Ebert,  204,  222. 

Effective  force,  48,  216. 

Electromagnetic,  energy,  6,  43,  212; 
inertia,  40,  212. 

Energetics,  2,  3,  6,  202. 

Energ>-,.configiu-ation,  187;  conserva- 
tion of,  4;  conversions  of,  45,  52; 
electromagnetic,  6,  43,  212;  flux  of, 
44,  181,  183,  212;  internal,  64; 
molecular  and  atomic,  51;  over- 
emphasis on,  3;  potential,  4,  199; 
storage  of,  6,  7. 


Energy-changes,  and  momentum, 
193,  194;  fictitious,  199. 

Energy  factors,  and  Lagrange  equa- 
tions, 186-187. 

Energy  forms,  and  mechanisms,  181. 

Energy  transfer,  and  Lagrange  equa- 
tions, 181. 

Envelope  solutions,  220. 

Equation  of  condition,  precession, 
170. 

Equation  of  impulse,  44,  46. 

Equation  of  motion,  44,  48;  and 
degrees  of  freedom,  183;  character 
of,  53,  revision  of,  52. 

Equation  of  work,  44,  46. 

Equations,  and  identities,  23,  44,  48; 
Euler's,  155-166,  180,  192;  La- 
grange's, 8,  180,  222. 

Equilibriiun,  and  adjustment,  69, 
174,  216;  fictitious,  50. 

Equivalence,  208,  224;  of  particle,  28, 
64. 

Euler,  2,  34,  112,  114,  147,  155,  221; 
configuration  angles,  114;  dynami- 
cal equations,  155-166,  180,  192; 
geometrical  equations,  117. 

Euler  equations,  and  moments  of 
inertia,  162,  180;  and  principal 
axes,  157,  160,  162-163;  and  rota- 
tion, 155;  and  shift,  160-161,  164- 
165;  apply  to  rigid  body,  155; 
generalized  form,  162-165;  and 
Lagrange,  192. 

Experiment,  and  impressed  force,  52. 

Fictions,  in  dynamics,  8;  in  energy- 
changes,  199;  in  force,  41. 

Fictitious,  equilibrium,  50;  transla- 
tion, 28. 

Flux,  of  energy  and  momentum,  44, 
181,  183,  212. 


228 


Fundamental  Equations  of  Dynamics 


The  Numbers  refer  to  Pages 


Force,  activity  of,  35,  36,  201;  a  dis- 
tributed vector,  34,  a  fundamental 
quantity,  21-22;  and  fluxes,  45;  and 
momentum  change,  32  and  vari- 
able mass,  38,  40,  42;  effective,  48, 
216;  fictitious,  41;  generalized,  7, 
184,  223;  gyroscopic,  137;  im- 
pressed, 48,  216;  ponderomotive, 
13;  supplemented  by  force-moment, 
36,  63;  transmitted,  50. 

Force  elements,  and  rotation,  66,  168; 
and  transfer-forces,  46. 

Force-moment,  34^36,  210;  a  funda- 
mental quantity,  21;  and  angular 
acceleration,  68,  69,  71-72;  and 
center  of  mass,  60,  63;  and  pre- 
cession, 171-172;  and  principal 
axes,  74;  and  rigid  solids,  60,  61; 
and  rotation,  66-68,  71,  74;  and 
rotation-axis,  167-168;  and  rota- 
tion-vector, 68;  and  shift,  106,  161; 
and  tangent-normal,  166-167;  a 
resultant,  35-36,  210;  directive,  68; 
disturbing  precession,  174;  supple- 
ments force,  36,  63. 

Forces,  and  degrees  of  freedom,  184; 
directional,  14Q;  equivalent  through 
work,  184;  gene/alized,  183-186; 
lost,  50. 

Fourier.  221. 

Free  vectors,  and  shift,  100-104. 

Fundamental  groups,  relation  of,  22, 
43-44. 

Fundamental  quantities.  21,  63,  65, 
113,  140,  154;  and  invariance,  83; 
and  reference-frames,  23,  24,  trans- 
fer for,  24. 

Gauss,  5. 

Generalized,  coordinates,  180,  182; 
Euler's  equations,  162-165;  forces, 


7,    183-186;   momentum,    7,    182, 

190;  velocity,  182,  186-187,  191. 
Geometrical,     conditions,    52,     213; 

equations,  117,  191. 
Gravitation,  and  energy,  51. 
Gray,  204. 
Gyroscope,  139,  163,  169,  193,  207, 

221;  diverts  energy,  179;  weak  or 

strong,  177. 
Gyroscopic,  forces,  137. 

Hamilton,  2,  8,  180,  200. 

Hansen,  141,  218,  220. 

Hayward.  217. 

Heaviside,  201,  204,  210,  212,  222, 

223. 
Helm,  202,  205,  212,  213. 
Holonomous,  223. 
Homogeneous,  body,   31;  functions, 

182,  223. 
Huyghens,  4. 

Ideal  coordinates,  141-147;  and  ac- 
celeration, 144;  and  polar,  142; 
and  shift,  143,  147;  and  tangent- 
normal,  151-152;  and  velocity,  142. 

Identities,  and  equalities,  23,  44,  48. 

Identity  and  continuity,  34. 

Ignoration,  of  coordinates,  194-200, 
224. 

Ignored  force,  and  variable  mass, 
40-41. 

Imaginary,  precession,  175,  177,  179. 

Impact,  42,  212. 

Impressed  force,  44,  48,  216;  and 
center  of  mass,  48-49;  and  rigid 
solids,  55,  62-63;  and  rotation, 
62-63;  and  translation,  62-63;  ex- 
perimentally known,  52. 

Independence,  of  coordinates,  182; 
of  rotation  and  translation.  63. 


Index 


229 


The  Numbers  refer  to  Pages 


Indeterminate   multipliers,   53,   213. 

Indi\'iduality,  of  masses  and  points, 
34,  81. 

Inertia,  6,  7,  16;  variable,  211. 

Integration,  and  shift,  126,  153-154. 

Internal,  actions  and  energy,  42,  51- 
52,  55,  64,  205. 

Interpretation,  mechanical,  12,  13, 
43,  187. 

Invariance,  and  center  of  mass,  83; 
of  acceleration,  83,  90;  of  funda- 
mental quantities,  83;  of  moments 
of  inertia,  160;  of  radius-vector, 
80;  of  velocity,  82,  90. 

Invariant,  frame-groups,  84. 

Inverse  square,  law  of,  5. 

Jacobi,  218,  220. 

Kinematics,  and  dynamics,  9,  12,  13, 
54,  72,  165;  and  transfer,  77. 

Kinetic  energj-,  a  flux,  44;  a  funda- 
mental quantity,  21;  analogues  of, 
7;  and  generalized  velocity,  182, 
191;  and  gravitation,  51;  and  inter- 
nal actions,  51-52;  and  particle,  28, 
29;  and  principal  axes,  74,  157;  and 
rigid  solids,  55,  60;  and  rotation, 
63,  65,  71,  74;  and  translation,  28, 
63:  a  scalar  product,  22,  25;  con- 
vection of,  45;  diversion  of,  64, 
179;  supplements  mean  values,  29, 
30. 

Kinetic  potential,  200,  224. 

Klein  and  Sommerfeld,  216,  220,  221. 

Konigsberger,  224. 

Lagrange,  2,  7,  8,  180,  184,  213,  222. 

Lagrange  equations,  180-200,  204, 
223;  and  energy-  factors,  186-187; 
and  Euler's,  180,  192;  and  polar 
components,    188;    and    standard 


frame,  187;  and  tangent-normal, 
187;  are  scalar,  184,  186,  222;  in- 
clude constraints,  187. 

Larmor,  205,  224. 

Latency,  of  momentum  and  energj', 
7,  45. 

Law,  of  inverse  square,  5;  of  inertia- 
change,  40. 

Laws,  of  motion,  4,  32-33,  50,  201. 

Localized  vectors,  22,  26,  36;  and 
shift,  104-106. 

Lorentz,  202,  207,  209,  213. 

Lost  forces,  50. 

Mach,  205,  206,  212. 

Maclaurin,  112. 

Mass,  and  volume-integral,  30,  31 
as  quotient,  3,  40;  constancy  of,  25 
continuity  of,  16;  generalized,  6 
variable,  38. 

Mass  average,  and  precision,  49;  of 
acceleration,  37;  of  velocity,  29. 

Mass  constancy,  33. 

Mass-summation,  22;  differentiated, 
24,  32-33. 

Mathematics,  and  dynamics,  36-37, 
113,  137,  139,  174-175,  208,  216, 
220;  simplifies,  17. 

Mattioli,  213. 

Maxwell,  6,  203,  224. 

Mean  values,  210;  residues  from,  30, 
36,  60-63. 

Mean  vector,  28. 

Mechanical  models,  12,  13,  43,  181, 
187,  213,  224. 

Molecular  energy,  51. 

Moment  of  momentum,  22,  27,  210; 
a  fundamental  quantity,  21;  a 
localized  vector,  22,  27;  and  par- 
ticle, 29;  and  precession,  171;  and 
principal  axes,  73;  and  rigid  solids, 


230 


Fundamental  Equations  of  Dynamics 


The  Numbers  refer  to  Pages 


55,  60, 73, 156;  and  rotation- vector, 
27,  68,  69;  and  shift,  106;  and  trans- 
lation, 28;  and  volume-integral, 
30,  31;  supplements  mean  values, 
29,  30. 

Moments  of  inertia,  and  Euler  equa- 
tions, 162,  180;  in  variance  of,  160. 
162. 

Momentum,  22;  a  distributed  vector, 
26;  a  flux,  44;  a  fundamental 
quantity,  21;  and  center  of  mass, 
29,  63;  and  generalized  velocity, 
182;  and  translation,  28;  and  vari- 
able energy,  193-194;  and  volume- 
integral,  30-31;  convection  of,  45; 
generalized,  182,  190;  invented  by 
Newton,  4;  latency  of,  7,  45;  recti- 
fied, 153;  transformed,  45. 

Momentum  change,  and  force,  32, 
36,  46. 

Motion,  absolute,  9,  10;  equation  of, 
48;  relative  to  center  of  mass,  28, 
38,  215;  second  law  of,  32-33; 
third  law  of,  50,  201. 

Multipliers,  indeterminate,  53,  213. 

Newton,  4,  9,  32-33,  50,  201. 

Notation,  comparison-frame,  78;  prin- 
cipal axes,  157-158;  standard 
frame,  77-78. 

Oblique  coordinates,  116,  219. 
Orthogonal  axes,  adopted,  23. 
Ostwald,  202,  212. 

Parameters,  Lagrange's,  180. 

Partial  derivatives,  91-96,  125,  149, 
185,  190.  216,  218,  219. 

Particle,  28;  and  center  of  mass,  37- 
38;  and  energy,  29;  and  moment  of 
momentum,   29;   and   polar   com- 


ponents, 139-140:  and  rigid  solid, 
54;  and  tangent-normal,  154;  equi- 
valence of,  64. 

Phenomenology,  202. 

Poincare,  81,  202,  206,  213. 

Points,  individualized,  81;  motion  of, 
81-82. 

Polar  components,  140;  and  ideal  co- 
ordinates, 142;  and  Lagrange  equa- 
tions, 188;  and  pure  rotation,  138- 
139;  and  superposition,  136;  and 
tangent-normal,  148,  149;  uni- 
planar,  136. 

Polar  coordinates,  130-135. 

Polar  velocity,  and  shift,  133. 

Ponderomotive  force,  5,  13. 

Position   coordinates,   auxiliary,   81. 

Potential,  5;  energy,  4,  7,  51,  199; 
kinetic,  200,  224, 

Power;  35,  36;  a  fundamental  quan- 
tity, 21;  and  center  of  mass,  60,  63; 
and  directive  action,  68,  140;  and 
rigid  solid,  60,  61,  63;  and  shift, 
140-141,  152-153;  and  variable 
mass,  39,  42. 

Power  equation,  201. 

Poynting,  213. 

Precession,  169-174;  condition  for, 
170;  imaginary,  175,  177,  179. 

Precision,  207,  210;  and  mass  average, 
49. 

Principal  axes,  73,  157,  160,  162-163; 
and  energy,  74;  and  Euler  equa- 
tions, 157,  160,  162;  and  force- 
moment,  74;  and  moment  of  mo- 
mentum, 73;  notation  for,  157- 
158. 

Principle,  d'Alembert's,  60;  Hamil" 
ton's,  8;  of  vis  viva,  4. 

Principles,  discovery  of,  18;  stability 
of,  8. 


Index 


231 


The  numbers  refer  to  Pages 


Projection,  of  angle- vector,  58,  115- 
116. 

Proximate  reference,  11. 

Pure  rotation,  59;  and  center  of  mass, 
64-65,  75;  and  constraints,  75;  and 
polar  compionents,  138-139. 

Quantity  of  motion,  32. 

Radius- vector,  in  variance  of,  80; 
mean,  28:  partial  derivative  of,  91- 
96;  prominence  of,  27,  36,  209,  210, 
214;  typical  character  of,  99. 

Reduction  factor,  190,  218,  224. 

Reference-axes,  orthogonal,  23. 

Reference-frame,  conceived  fixed,  23; 
postponed  choice  of,  24,  207;  proxi- 
mate, 11;  transfer  for,  76;  ulti- 
mate, 9,  10,  11,  88. 

Reference-frames,  configuration  of, 
78-79;  invariant  groups  of,  84. 

Regular  precession,  169-174;  and 
force-moment,  171-172;  and  mo- 
ment of  momentum,  171;  dynamics 
of,  171-174;  imaginary,  175,  177, 
179. 

Relativity,  4,  11,  202,  213. 

Representative  prarticle,  28. 

Resolution,  tangent-normal,  40,  147- 
154. 

Resultant  elements,  force,  34;  force- 
moment,  36,  210;  moment  of  mo- 
mentum, 27,  210;  momentum,  22. 

Revision,  of  physical  equations,  52. 

Rigid  dynamics,  and  Euler  equa- 
tions, 155,  221;  approximate,  53. 

Rigidity,  214,  215;  and  internal 
energ>^,  55;  of  ultimate  parts,  54. 

Rigid  solid,  53;  and  center  of  mass, 
55;  and  Euler  equations,  155;  and 
force-moment,    60,    61;    and    im- 


pressed force,  55,  62-63;  and  mo- 
ment of  momentum,  55,  60,  73, 
156;  and  particle,  54;  and  power, 
60,  61,  63;  and  rotation,  55,  57,  58, 
63,  215;  angular  velocity  of,  57; 
general  motion  of,  63;  structure  of, 
53,  55,  62. 

Robb,  207. 

Rotation,  55-57,  215,  and  accelera- 
tion, 65-66;  and  center  of  mass,  55, 
57,  60;  and  energj-,  63,  65,  71,  74: 
and  Euler  equations,  155;  and 
force-moment,  66-67,  68,  71,  74; 
and  impressed  force,  62-63;  and 
uniplanar  motion,  72,  167;  and 
velocity,  57,  59;  of  rigid  solid,  55, 
57,  58,  63. 

Rotational  stability,  169,  175-179; 
condition  for,  176,  178. 

Rotation-axis,  and  force,  66,  168; 
and  force-moment,  167-168, 

Rotations,  superposition  of,  59. 

Rotation- vector,  57,  208,  214;  and 
configuration  angles,  117-123;  and 
force-moment,  68;  and  shift,  123, 
124,  126-131;  and  standard  frame, 
58-59;  divergence  from  moment  of 
momentum,  27,  67-70. 

Routh,  214,  217,  221. 

Scalar  equations:  cartesian,  107-111; 
Lagrange's,  184,  186;  standard 
frame,  112,  187. 

Shift,  94,  97,  216,  217,  218;  and  ac- 
celeration, 150,  152;  and  angular 
acceleration,  126-131,  160,  164; 
and  cartesian  axes,  107-111;  and 
Euler  equations,  160-161,  164;  and 
force-moment,  106,  161;  and  free 
vectors,  100-104;  and  ideal  co- 
ordinates, 143,   147;  and  integra- 


232 


Fundamental  Equations  of  Dynamics 


The  numbers  refer  to  Pages 


tion,  126,  153-154;  and  localized 
vectors,  104-106;  and  moment  of 
momentum,  106;  and  motion  com- 
pared, 96-97,  104;  and  polar  ac- 
celerations, 134;  and  polar  veloci- 
ties, 133. 

Shift  rate,  97-98;  and  power,  140, 
152-153;  and  rotation-vector,  123, 
124,  126-131. 

Silberstein,  202,  222. 

Simplifications,  in  dynamics,  17,  54, 
205. 

Space  curves,  acceleration  in,  151- 
152. 

Stability,  condition  for,  176,  178;  of 
principles,  2,  8,  15;  rotational,  169, 
175-179. 

Standard  frame,  and  fundamental 
quantities,  113;  and  Lagrange 
equations,  187;  and  rotation-vec- 
tor, 58-59;  arbitrary  choice  of,  78; 
as  coordinate  system,  112;  nota- 
tion for,  77-78. 

Storage  of  energy,  6,  7,  64. 

Summation,  continuous  or  discrete, 
23. 

Superfluous  coordinates,  183,  190, 
191,  223. 

Superposition,  59,  88,  216;  failure  of, 
136;  of  rotation  and  translation, 
63. 

System,  conservative,  5, 7;  dynamical, 
16;  internal  connections  of,  49-50; 
of  bodies,  16. 

Tait,  201. 

Tangent-normal,  40,  147;  and  ac- 
celeration, 148;  and  force-moment, 
166;  and  fundamental  quantities, 
154;  and  ideal  coordinates,  151- 
152;  and  Lagrange  equations,  187; 


and  polar  components,  148,  149; 
and  velocity,  147;  as  prototype, 
149. 

Tensors,  derivatives  of,  93,  96,  102, 
154;  groups  of,  92. 

Thomson  and  Tait,  201,  203,  214, 
218,  221,  222,  224. 

Time-derivative,  of  geometrical  equa- 
tions, 191. 

Time  functions,  two  classes  of,  98, 
218,  220. 

Timeless  comparisons,  81. 

Total  and  partial  derivatives,  91-96, 
125,  128,  219. 

Total  force,  34;  and  center  of  mass, 
37. 

Transfer:  angular  acceleration,  124; 
angular  velocity,  124;  reference- 
frame,  24,  76,  77. 

Transfer- force,  45;  a  distributed  vec- 
tor, 47;  and  local  resultants,  46;  as 
constraints,  47. 

Transformation,  of  momentimn,  45. 

Translation,  27,  28;  and  center  of 
mass,  37-38,  63,  215;  and  energy, 
28,  63,  and  impressed  force,  62-63; 
and  rigid  solid,  55,  63;  and  rotation, 
63,  215. 

Transmission  of  force,  50. 

Ultimate,  reference,  9,   10,  11,  205; 

rigidity,  54. 
Uniplanar,    acceleration,    136,    150; 

rotation,  72,  167. 

Variable  mass,  38-42,  211;  and  ig- 
nored force,  40-41 ;  and  summation, 
24-25. 

Vector  algebra,  13,  208. 

Vectors,   descriptive,   137,   141;   dis- 


Index 


233 


The  Numbers  refer  to  Pages 


tributed,  26,  34,  45;  shift  for,  100- 
106. 
V^elocity,  82;  and  center  of  mass,  29, 
55,  57,  and  ideal  coordinates,  142- 
144;  and  rotation,  57,  59;  and  tan- 
gent-normal, 147,  152;  angular,  57; 
generalized,  182,  186-187,  223;  in- 
variance  of,  83,  90;  mass-average 
of,  29;  partial  derivative  of,  149; 
polar  components  of,  132,  133,  136; 


relative  to  center  of  mass,  55,  57; 

transfer  for,  85-88;  virtual,  50. 
Virtual,  velocity,  50;  work,  7,  50. 
Vis  viva,  principle  of,  4. 
Volume  integrals,  30,  31. 

Whittaker,  224. 
Work,  virtual,  7,  50. 
Work  equivalence,  and  force,  46,  183- 
184,  223. 


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